the topology of real projective algebraic varieties
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THE TOPOLOGY OF REAL PROJECTIVE ALGEBRAIC VARIETIES
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Russian Math. Surveys 29:4 (1974), 1-79From Uspekhi Mat. Nauk 29:4 (1974), 3-79
Dedicated to the memory ofIvan Georgievich Petrovskii
THE TOPOLOGY OFREAL PROJECTIVE ALGEBRAIC VARIETIES
D. A. Gudkov
This article is a survey of the results on Hubert's 16th problem from 1876 to the present.1. HARNACK'S THEOREM. The number of branches of a non-singular curve of order m in R/>2
does not exceed ̂ {rn — 1) (m — 2) + 1. Harnack's method for constructing curves with the greatest numberof branches (M-curves). The methods of Hubert, Brusotti, and Wiman for constructing Λί-curves. M-curveson a quadric.
2. The concepts of roughness and degrees of non-roughness. Brusotti's theorem on the independenceof the simplifications (small corrections) of simple double points and cusps.
3. The theorems of Petrovskii and Oleinik on algebraic curves and surfaces. The generalization due toKharlamov.
4. Sextic curves in RP2. Cubic and quartic surfaces in RP3.5. An oval of a non-singular curve is said to be odd if it lies inside an odd number of other ovals, and
even otherwise. Let A be an M-curve of even order m, and suppose that the number of its even (odd)ovals is Ρ (L). Then Ρ — L = (m/2)2 mod 8. This congruence was proved by Arnol'd mod 4 and byRokhlin mod 8. Broad generalizations were given by Rokhlin and others.
6. Some conjectures.
Contents
§ 1. Introduction 2§2. The change in the topology of a curve in RP 2 under a change
in the coefficients of the curve 4§3. Harnack's theorem and the construction of Λί-curves 16§4. The methods of Hilbert and Brusotti for the construction of
Af-curves 19§5. Λί-curves on a quadric 24§6. The theorems of Petrovskii and Oleinik 28§7. Sextic Curves in RP 2 40§8. Amol'd's theorems 45§9. Harnack's theorem for varieties and Rokhlin's theorems . . . . 48§ 10. Periodicity of the Euler characteristic of an (M - l)-variety . . 58§11. Cubic and quartic surfaces in RP 3 62§ 12. Conclusion 67References 71
2 D. A. Gudkov
§1. Introduction
The study of the topology of real algebraic varieties is classical. Coniesand quadrics were studied in antiquity. Descartes, Newton, Euler, Plikker,and other mathematicians, made their contribution to the study of curvesof higher orders. Hubert was the first to pose questions on the topology ofnon-singular algebraic varieties. I recall that Hubert's 16th problem consistsof two parts: the first part, which studies the mutual disposition of themaximal number (in the sense of Harnack) of separate branches of analgebraic curve, and also the "corresponding investigation" for non-singularreal algebraic varieties; and the second part, which poses the question ofthe maximal number and disposition of the limit cycles of the differential
equation ^ = "|*' , , where Pn and Qn are polynomials of degree n. In
the first part Hubert refers to difficult particular problems in the topologyof sextic curves and quartic surfaces; see [15], [119].
The present article surveys the results on the first part of Hubert's 16thproblem. In addition, some results on algebraic curves with singular pointsare included.
The survey covers the time from 1876 to the present. This time interval,in my opinion, can naturally be divided into three periods.
FIRST PERIOD (1876-1933). The application of the "small parameter"method is characteristic of the majority of papers of this period. In 1876,Harnack [7], using a theorem of Bezout, proved that a non-singular curveof order m in RP 2 can not have more than \{m - \){m — 2) + 1 separatebranches (Harnack's theorem).
In the same paper Harnack proved by the "small parameter" method thatcurves with this maximum number of branches C/W-curves) exist. In 1891,Hilbert [10] posed the question of the mutual disposition of the ovals ofM-curves, gave another method of constructing M-curves, and proved theanalogue of Harnack's theorem for curves on a quadric in RP 3 . Othermethods were given by Brusotti [22]- [28] , [30], [31] in 1910-1917, andby Wiman [35] in 1923. I mention two more results of this period. In1892, Klein and Hurwitz [11], [14] gave a new proof of Harnack's theoremby means of Riemann surfaces. In 1921, Brusotti [32] proved a theorem onthe independence of the simplifications of the simple double points of aplane curve. He used the properties of the canonical linear series on acurve and a theorem of Max Noether.
SECOND PERIOD(1933-1971). The application of a continuous non-small change in the coefficients of a curve, surface, etc., is characteristicof the papers of this period. The papers of Petrovskii and Oleinik occupya central position. In 1933-1938, Petrovskii [50], [56] proved that if/ = 0 is the equation of a non-singular curve of even order in RP 2 , if theset B+(f> 0) is orientable, and if χ(Β+) is the Euler characteristic of B+,
The topology of real projective algebraic varieties 3
then \{3m2-6m) < χ(Β+) < | ( 3 m 2 - 6 m ) + 1 (Petrovskii's theorem). Thisproposition had been stated by Ragsdale [17] in 1906 as a conjecture.The results of [56] were generalized by Petrovskii and Oleinik in 1949 tohypersurfaces [80], [82], [91], and by Oleinik in 1950 to space curves[88] —[90]. Petrovskii and Oleinik used Morse theory, (in particular, thecontinuous non-small change in the coefficients of a curve, surface, etc.)and the Euler-Jacobi theorem for the roots of systems of algebraic equations.In 1949-1951 Oleinik [81], [92], using the same methods, gave estimatesfor the sum of the Betti numbers of a real algebraic hypersurface. Analogousestimates were given by Milnor [114] in 1964 and by Thorn [115] in 1965with the help of Morse theory. In [115], Thorn, using Smith's theory, gavean estimate for the sum of the Betti numbers of a real hypersurface. Thisestimate was a generalization of Harnack's theorem, a fact that Thorn himselfdid not notice; see [115], [126], [127].
In 1967, I was able to solve the problem of the topological classificationof the disposition of the ovals of a non-singular sextic curve in R P 2 . Iemphasize that I made essential use of the papers of Rohn [23], [25]and the ideas of Andronov on the application of the concepts of roughnessand degrees of non-roughness in the study of the change in the topology ofa curve (surface) under a continuous non-small change in the coefficientsof the curve (surface). The solution of the problem of sextic curves enabledme to formulate the following conjecture [123]: if / = 0 is the equationof an M-curve of even order m in RP 2 and if the set B+ (/ > 0) isorientable, then
χ{Β+) == (m/2)2 mod 8.
THIRD PERIOD (1971 to the present). This period is characterized bythe fact that the investigations on the topology of real algebraic varietieshave flowed into a common current with the investigations on differentialtopology. The merit for this is due primarily to Arnol'd [124] andRokhlin [126].
In 1971, Arnol'd [124] discovered profound connections between thetopological disposition of the ovals of an M-curve and general facts ofdifferential topology and created a fruitful method of studying M-curves.In [124], the congruence χ(Β+) = (m/2)2 mod 4 is proved for M-curves ofeven order, a new proof of Petrovskii's theorem is given, and new approachesto the study of the mutual disposition of the ovals of plane curves aresuggested.
Rokhlin [125], [126] first solved the problem of generalizing Harnack'stheorem; namely, he proved two variants of such a generalization (thetheorems of Harnack—Thorn and Harnack—Rokhlin). In this context heobtained two generalizations of the concept of an M-curve.
M-varieties and M-pairs. Rokhlin proved congruences for M-varieties andM-pairs, analogous to the congruence x(B+) = (m/2)2 mod 8 for M-curves
4 D. A. Gudkov
(in particular, this very congruence). Rokhlin [125], [126], [130] usedthe apparatus of differential topology extensively for the study of thetopology of real algebraic varieties.
Rokhlin's work was continued by Kharlamov [129] and Gudkov andKrakhnov [128].
Finally, Kharlamov [131] proved inequalities generalizing the inequalitiesof Petrovskii and Oleinik in [56], [82], [90].
Of the unsolved problems I mention just one. The greatest difficultiesarise in the construction of all essential Μ -curves of a given order m. Sofar this has been done only for m < 6. In the survey an attempt is madeto formulate some new conjectures on the disposition of the ovals of non-singular plane curves. There are also other conjectures.
The foundation for these conjectures is extensive experimental materialgiven by well known methods of constructing non-singular curves.
I would like to express my gratitude to E. A. Leontovich-Andronova,O. A. Oleinik, V. I. Arnol'd, V. A. Rokhlin, S. P. Novikov, and V. M.Kharlamov for discussions and for their interest in my work.
§2. The change in the topology of a curve in RP2 under achange in the coefficients of the curve
This section begins with the definition of certain concepts. Following anidea of Arnol'd, we introduce and analyze the concepts of roughness anddegrees of non-roughness of a curve in some regular domain of R P 2 . Thenwe study certain simplifications of a curve in a regular neighbourhood of asingular point of it and prove Brusotti's theorem on the independence ofthe simplifications of the singular points of a curve having only simpledouble points and cusps.
Finally, we present some theorems on the variability of the singularpoints of a simple curve and formulate the marking method (the "smallparameter" method).
1. A real algebraic curve Cm (of the mth order) in R P 2 is defined byan equation
Cm s ν Aaiixvxpl = 0,
where m is a fixed natural number; α, β, and y are non-negative integers;the Ααβ are real numbers, at least one of which is non-zero; [x0: χγ: χ2]are the coordinates of a point in R P 2 .
The set of all real curves Cm (of a fixed order m) is the real projectivespace RP^, where TV = m(m + 3)/2; see [136*] ,1 [102]. We call it thespace of curves of the wth order.
1 Auxiliary references are marked by an asterisk in the list of references.
The topology of real projective algebraic varieties 5
We introduce a distance between two points x[x0: x1: x2] and
:yi'.y2] in the complex projective plane C P 2 by the formula
(1) Ρ (*. y) =arccos j^ '-ltf '
where (x · y) = χο>Ό + XiJi + X2^2 a n d \χ Ι = \Λ χ ' χ ) ( s e e [ 137*]We denote the ω-neighbourhood of ζ £ CP2 by Ω*(ζ, ω). As a modelof RP 2 we take the sphere
(2) xl + A + x\ = 1with diametrically opposite points identified (the projective sphere). Wedenote the ω-neighbourhood of ζ in RP2 by Ω(ζ, ω). The distance (1) onthe projective sphere is the length of the smaller arc of the great circlepassing through χ and y. For our scheme to hold we take the upper half-sphere x 0 > 0 of (2) and project it orthogonally onto the diskx\ + x\ <Ξ 1, x 0 = 0. The resulting disk, with diametrically opposite cir-cumference points identified, is called the projective disk.
We introduce a distance in RP^ by a formula analogous to (1), andthe ε-neighbourhood of F £ R P ^ by U(F, ε).
Let z[z0: zx:z2\ £ RP 2 be a point1 of a curve F £ RP^, and supposethat there are no singular points of F except (perhaps) ζ in a sufficientlysmall neighbourhood Ω*(ζ, ω). It is known (Puiseux' lemma) that thereexists an ω 0 > 0 and a natural number /, such that any point of F inΩ*(ζ, ω 0 ) (except ζ) is obtained for one and only one value tv in oneand only one of the / expansions ($PV)
where av Φ 0, the d are constants, and tv is an independent variable.The expansion (&PV) is called an irreducible parametrization of F (or of a
component of F if F decomposes). A class of equivalent irreducible para-metrizations (see [136*] is called a branch &PV of F with centreat z. As a rule, one of the representatives of this class, that is, (^v)> IS
also called a branch, as well as the set of points of F in Ω*(ζ, ω 0 )obtained under the appropriate change of tv. A branch <iPv is said to bereal if there exists a parametrization (£Ρν) in which all_the constants av,and a1 are real. Obviously, in a real neighbourhood Ω(ζ, ω 0 ) only realbranches are represented.
A complete real analytic continuation of a real branch ( # v ) of F in RP 2
is called a complete real branch of F. A complete real branch 7 of F issaid to be even (odd) if there exists a real line L that intersects 7 in aneven (odd) number of real points each of which is non-singular for F andnot a point of tangency of 7 and L. Every complete real branch of F is
1 Sometimes a point F e RP is called, by an abus de language, a curve F e RP .
6 D. A. Gudkov
either even or odd. A complete even branch 7 of F is called an oval ifthere are no singular points of F on 7.
2. We note that it is not sufficient to watch only for the invariance orchange of the topological disposition of a curve (or its parts) in RP 2 asits coefficients change continuously. For example, if a curve F has only onesingular point, a cusp, in R P 2 , then for any ε > 0 one can find aΦ G U(F, ε ) without singular points that is isotopic to F in RP 2 . How-ever, F and Φ must be distinguished. Therefore, it is necessary to specifyexactly what is meant by the "invariance" or "change" of a curve as itscoefficients change continuously (see [116]).
Let G be a domain in RP 2 . The boundary1 of G consists of a finitenumber of pairwise disjoint metric circles (in the metric (1)). We call sucha domain regular. We say that a curve F Ε RPN is disposed regularly in Gif it has a finite number of singular points in G, is not tangent to theboundary of G, and has no singular points on the boundary. In particular,RP 2 is a regular domain, and any curve without multiple components isdisposed regularly in RP 2 .
By analogy with the qualitative theory of differential equations it is nothard to formulate the concepts of roughness and degree of non-roughnessfor curves F £ RPN (m is fixed) disposed regularly in a regular domain G.
A curve F has a zero degree of non-roughness (is rough) in G if thereexists an ε0 > 0, such that any Φ ς U(F, ε0) is isotopic to F in G. Acurve F is of the (s + l)st degree of non-roughness in G if, firstly, forevery ε > 0 there exists a Φο 6 U{F, ε) of the 5th degree of non-roughness in G and not isotopic to F in G, and, secondly, there exists anε0 > 0 such that any curve Η 6 U(F. ε0) that is not of degree of non-
roughness k < s in G is isotopic to F in G (s = 0, 1, . . . ).2
We note further that two curves of the mth order disposed regularly inG, isotopic, and having the same degree of non-roughness in G, canessentially be distinguished by determining what curves can be obtainedfrom them under arbitrarily small changes of their coefficients.' For example,two curves F and Φ of the fourth order, each consisting of one completeeven branch with a cusp in RP 2 and distinguished by the fact that theacute cusps of F are directed toward the interior region of this branch, andthose of Φ toward the exterior. These curves are isotopic in R P 2 and areof the second degree of non-roughness in R P 2 . However, non-isotopiccurves with a simple double point can be obtained from F and Φ by smallcorrections. We focus in on this more detailed distinction between curvesby the concept of type of a curve F in a domain G.
1 One can treat more complicated boundaries [116].The idea of using these concepts for investigating the topology of algebraic curves is due toAndronov. It can be proved that an algebraic curve having only simple double points (that is, doublepoints with distinct tangents) has a definite degree of non-roughness in RP2. This can also be provedfor certain other classes of curves. It is still unproved that a general curve has a degree of non-roughness;see [116].
The topology of real projective algebraic varieties η
Two curves F and Φ (in RPN) have the same type of zero degree (roughtype) in G if F and Φ are disposed regularly in G, are of degree zero ofnon-roughness in G, and are isotopic in G.
Two curves F and Φ have the same type of the (s + \)st degree(s > 0) in G if, firstly, F and Φ are of the (s + l)st degree of non-roughness in G, secondly, F and Φ are isotopic in G, and, thirdly, for anysufficiently small ε > 0 in the neighbourhoods U(F, ε) and ί/(Φ, ε) thereexist curves of the same types of the sth degree in G (s = 0, 1 , 2 , . . . ) .
We can now define the concepts of "invariance" and "change" that werequire.
We say that the type of F in G is preserved along a set Μ C RPN if Μis linearly connected, F £ M, and every curve Η £ Μ has the same type asF in G.
The type of the sth degree of F in G is simplified along Μ C RPN to atype of the kth degree (k < s) if, firstly, F is a limit point of M, and forany ε0 > 0 there is an ε > 0 (ε < ε0) such that Μ (Ί U{F, ε) islinearly connected in Μ Π U(F, ε0) and, secondly, the curves Η &Μ have thesame type of the kth degree in G.
If the type of F (or the curve F) is simplified to the type of Η Ε Μ(to the curve Η G M) (denoted by F -*• Η £ M), then we also say that Fis obtained by a complication of Η G Μ (denoted by Η => Λί). But if F isobtained by a complication oi Η Ε Mx and if F is simplified along M 2,then we say that the curves Η £ A^ U M2 bifurcate through F in G.
3. It is natural to begin the study of the simplifications of a curveF G RPN in some domain with that of the simplifications of F in a"sufficiently small" neighbourhood Ω(ζ, ω) of an isolated singular point1
ζ of F (that is, one not lying on a multiple component of F).Let us define the concept of "sufficiently small" neighbourhood more
precisely. Let ζ be a real isolated singular (or ordinary) point of F G RPN.We draw a unit circle Ε with centre at z. We denote by Lt(\ </</z) allthe real tangents to F at ζ (Fig. 1) and by bh ct, 1 < / < h, the pointsof intersection of Lt with E. It is obvious that there exists a numberΔ > 0 such that the closures of the neighbourhoods Δ(6,·) and A(c;)(1 < / < h) on Ε are pairwise disjoint. We set
E\ U
It is not hard to show that there exists a neighbourhood Q, = Q,(z,cjthat, firstly, the circle with centre at ζ and radius r, 0 < τ < ω, intersects eachreal branch aPv with centre at ζ in two and only two points withouttangency (we recall that the "branch aP v" is a local concept); secondly,each of these ^ v is projected from ζ into Ε - Q on E; thirdly, any lineL{q) through q G Q is not tangent to F in Ω and intersects F in this
A vast body of literature is devoted to the study of the singular points of algebraic curves. I refer only
t o a f e w works: [1] , [8] , [117].
8 D. A. Gudkov
neighbourhood in a number of points not greater (counting multiplicities)than the order of z. We call such a neighbourhood Ώ,-regular and keep allthe notation introduced above (L,·, bt, ch Ε, Δ, Q, . . . ). In addition, wedenote by av the points of intersection of F with the boundary of Ω. It isobvious that there exists a δ > 0 such that the closures of theδ-neighbourhoods δ(αν.) on the boundary of Ω are pairwise disjoint andeach δ(αν) is projected from ζ into Ε - Q on Ε (see Fig. 1).
Fig. 1
4. For our investigation of the simplifications of F in a regular neigh-bourhood Ω(ζ, ω) of a singular point ζ it is useful to introduce the follow-ing projective invariant of ζ on F (see [113]).
Let ζ be an isolated singular or ordinary point of F G RPN, ζ G CP2.We take a point q not lying on the tangent to F at ζ and let U(F, g) bethe first polar of F with respect to q. We denote the number of intersectionsof F and Π (F, q) at ζ by κ(ζ) and call it the class of z. The class κ(ζ)does not depend on the choice of q and is a projective invariant. If ζ doesnot lie on the line x0 = 0, if χ = Χι/χ0, y =
ΧΪΙ*Ο, and if the line Oy isnot parallel to the tangent to F at z, then κ(ζ) is equal to the number ofintersections of F(x, y) and Fy(x, y) at z. Hence we easily obtain thefollowing propositions:1
LEMMA ON THE CLASS OF A POINT. Let Ω be a regular neighbour-hood of a point ζ of F G RF^. Then there exists an ε0 > 0 such that,firstly, every curve Φ ζ U(F, ε0) is disposed regularly in Ω; secondly, ifav are the points of intersection of Φ with the boundary of Ω, thenav G δ(αν) for all av; thirdly, if Φ has the singular points zl, z2, . . . , zk
in Ω, then h
S1 Proofs of the four following lemmas can be found in [116] and [120].
The topology of real projective algebraic varieties 9
where the κ(ζ(·) are the classes of the zx for Φ, m"(q) is the number ofreal tangents from any q G Q to Φ in Ω, and κ(ζ) is the class of ζ for F.
LEMMA ON ISOTOPY. Let G be a regular domain, F G RPN, and supposethat F is disposed regularly in G. Let z= (1 < / < n) be all the singularpoints of F in G and Ω· regular pairwise disjoint neighbourhoods of theZ:, where the Ω.- do not intersect the boundary of G. Then there exists anε0 > 0 such that every curve Φ ζ U{F,E0) having a singular point z- in
each Ω.- for which κ(ζ-) = κ(ζ7-) and ord (Z) = ord (z·) (1 < / ' < « ) isisotopic to F in G.
It is not hard to show that κ(ζ) = 0 for any ordinary point of a curve,κ(ζ) = 2 for a simple double point, and κ(ζ) = 3 for a cusp.1
Let F G RPN and suppose that F is disposed regularly in a regulardomain G. For F to be rough in G it is necessary and sufficient that Fhas no singular points in G. For F to be of the first degree of non-roughness in G it is necessary and sufficient that F has only one simpledouble point in G. Similarly, for non-roughness of the second degree it isnecessary and sufficient that F has either two simple double points or onecusp in G.
LEMMAONTHE SIMPLIFICATIONS OF THE SIMPLEST SINGULARPOINTS. Let ζ be a real isolated singular point of F G RP A and Ω a regularneighbourhood of z, and suppose that ε0 > 0 satisfies the assertions ofthe lemma on the class of a point and that Φ £ U(F, ^a).
1°. If ζ is a simple double isolated point of F, then Φ can have onlyone of the three types A, B, and C of the first row of Fig. 2 in Ω.
2°. // ζ is a node of F, then Φ can have only one of the three typesA, B, and C of the second row of Fig. 2 in Ω.
3°. // ζ is a cusp of F and Φ has a singular point in Ω, then Φ can haveonly one of the three types A, B, and C of the third row of Fig. 2 in Ω.
Note that the curves Β and C in the second row of Fig. 2 are isotopicin Ω and hence have the same rough type in Ω. It can happen that thereexists a linearly connected set Μ containing Β and C and such that allΗ G Μ have the same type in Ω, but that for some ε0 > 0 and anyε > 0 (ε ^ ε0) the set Μ Π U(F, ε ) is not linearly connected inΜ Π U(F, ε0). We note that even for a simple triple point one cannotcompute by these methods all the possible simplifications of a curve in aregular neighbourhood of this singular point.
5. We introduce some more integral characteristics of a singular point.Suppose that F G RPN has an isolated singular or ordinary point
ζ G CP2. By the genus of z, which we denote by g(z), we mean thenumber
A double point is called a cusp (or spike) if it is the centre of the unique branch <P of order 2 andclass 1. The definition of the order and class of a branch can be found in [10], 131. A real simpledouble point ζ is said to be isolated if both branches,3a
1 and .Ψ'2, with centre at ζ are imaginary anda node \i{Px and^?s
2 are real.
10 D. A. Gudkov
where aPv is a branch with centre at ζ, γ(ίΡν) is its order, and the sum istaken over all branches with centre at z.1
LEMMA ON THE GENUS. Suppose that an indecomposable curve F hasorder m and genus p. Then
where the sum is taken over all the singular points of F.This lemma makes it possible to give an upper estimate for the logically
possible collections of singular points that a curve may have.The following conjecture is likely to hold:CONJECTURE ON THE GENUS OF A SINGULAR POINT. Suppose
that an indecomposable curve F of order m has a singular point ζ G CP 2
and that ω 0 > 0 is sufficiently small. Then there exists an ε0 > 0, suchthat for an Φ ζ U{F, ε0) having the singular points z- £ Ω * ( ζ , ω 0 )(1 < / < / : ) the following inequality holds:
The invariantsx(z),g(z), . . . and their corresponding algebraic formulae are essentially known (see[136*]). In [116] these invariants are used to investigate the simplifications of singular points.
The topology of real protective algebraic varieties 11
where g(z:) is the genus of Zj and g(z) is the genus of z.We denote by h(z) the number of intersection points of F and the
Hessian of F (see [136*]) at a point ζ of F. If F has no linear or multiplecomponents, then
(3) Sfc(z,)<3m(m-2).j
Therefore, the numbers h{zf) may be used in a manner analogous to thegenus. The inconvenience of the h(z) arises from the fact that (3) is"insensitive" to the decomposition of a curve. In addition, h{z) is notdefined uniquely by the singularity at z. For example, if ζ is a node, thenh(z) > 6. There is equality only if the branches^ and 5s
2 with centre atζ have no points of inflection at z. This suggests that one should considerthe number h(z) = h(z) - J ] [κ($\) - 1], where κ(3\) is the class of ^ v ,
V
and the sum is taken over all branches with centre at z. It is not hard toform a table
Table 1
No.
1234
5
!
Nature of the point ζ of F
Ordinary pointSimple pointCuspPoint of osculationSimple point of order r
ord 2
1222
r
κ (2)
0234
r(r-i)
0112
y(r-D
h (z)
>o>6>8
>12
>3r(r-l)
It is likely that h(z) = 2κ(ζ) + 2g(z).We return to the subject of this section. Suppose that F € RPN has a
real singular point z1; that Ω! is a regular neighbourhood of z1 ; and thata simplification of zx (that is, a simplification of F in Ω0 occurs alongΜχ C RP^. We say that simplifications of zx are identical along Mx andM1 if there exists an Μ C RP^ along which a simplification of Zj occursand Mx U ^ C M.
Suppose that F S RP^ has two real singular points zx and z? and thatΩ! and Ω 2 are disjoint regular neighbourhoods of them. Suppose thatzx (z2) is simplified or preserved along M1 (or M2). We say that thesesimplifications or preservations are compatible if there exists a set Μ alongwhich a simplification ofz! (or ζ2) occurs that is identical with that alongMi (or Λί2)·
Finally, suppose that F G RP^ has two real singular points zx and z2 and
12 D. A. Gudkov
that fij and Ω 2
a r e regular neighbourhoods of them. Suppose that eachpossible simplification or preservation of the type of F in Ωγ is compatiblewith each possible simplification or preservation of the type of F in Ω2.Then we say that the simplifications of zx and z 2 are independent.
The independence of the simplifications of an arbitrary number of singu-lar points is defined similarly. If we are only interested in the simplificationsthat lower the degree of non-roughness by unity, then we speak (analogously)of the independence of the highest simplifications of singular points.
In 1921 Brusotti [32] proved the following theorem:BRUSOTTI'S THEOREM. Suppose that a curve F G RP^ has no multi-
ple components and that all its singular points are simple double points {asimple curve). Let z}-, 1 < / < k, be all the real singular points of F,andΩ· pairwise disjoint regular neighbourhoods of these points (1 < / < & ) .We arbitrarily mark each of the points with one of the symbols B, A, andC. Then there exists a continuous curve Μ C RPN along which each of thez- is simplified according to the marked symbol, namely, as indicated inFig. 2 of the lemma in §2.4.
In other words, the simplifications of the real singular points of a simplecurve are independent.
I was able to obtain the following generalization of Brusotti's theorem[111], [113], [116]:
THE GENERALIZED BRUSOTTI THEOREM. Suppose that a curveF = Fx 'F2 *. . . •Fh is of order m. Suppose that the component Fj {of F)does not decompose, is of order mj, and has k.- cusps, where
(4) ft,<3ro, — 1, 1 </</*.
The remaining singular points of F are simple double points. Then for allreal singular points of F the highest simplifications are independent (see thelemma on the simplifications of the simplest singular points).
I mention that these simplifications can be produced so that any pair ofconjugate complex simple double points of F is preserved or vanishes (aswe wish) and so that any pair of conjugate complex cusps is preserved orcarried into a pair of complex simple double points.
Let me describe the idea of the proof of the last theorem. For brevitywe assume that all the singular points of F are real. We denote byz° 1 < ν < δ, the simple double points of F and by d°, 1 < μ < k, thecusps of F. Suppose that none of these points lies on the line xQ = 0. Wedenote by Ωμ, 1 < ν < δ, and ϋμ, 1 < μ < k, regular neighbourhoods ofthe corresponding singular points, and suppose that they are all pairwisedisjoint.
a) We take a simple double point z°. It can be shown that all curves Φsufficiently close to F having a singular point (simple double point) in Ών
form an analytic surface SPz»~ of dimension TV - 1 (a sheet) passing
through F, defined by
The topology of real projective algebraic varieties 13
The tangent hyperplane RP^f 1 to (5) at F has the equation
where [XQI>: ^ I ^ : * 2 I J are the coordinates of z°. The coefficients ^4a(3 arevariables.
b) We take a cusp d°. It can be shown that the set of curves sufficientlyclose to F having a singular point άμ in G' forms a hypersurface ί^,^Γ1
in RP^; the curves having a cusp in G., form an analytic surface &%~2
μ d\i
of dimension Ν - 2 that divides ^ i r 1 into two connected components+αΡ,ο and <9̂ do ( Φ 6 + ^ Λ Ο has an isolated point in G.,, and Φ 6 (̂ ,̂ο
α μ "μ «μ ** α μ
has a node) (Fig. 3). The tangent plane RPJ^ to &do~ at F has theequations μ
Η (dl) = 0, Ζ μ μ α β , 4 ) = F X 2 X 2 (ώ°μ) ΗΧ1 (4) - ·̂ι.χ2 (dji) HXi (ά°μ) = 0 .
Fig. 3
c) We assume that F does not decompose. Suppose that Fx^ (d°) Φ 0 ateach cusp. We form the system of linear equations
We denote by L the linear system of curves of order m that satisfy (6),L C CPN. We denote by r' the dimension of L. Let us compute r'. Wedenote by <^Vl 1» and <fFV) 2 the branches of F with centre at z° and by βthe branch of F with centre at d®. We consider the cycles (divisors)
Σ ( . , ) <ϊμ. C = Σv=l μ=1 μ=1
Let /n be the remainder of the linear series cut out by all the curves oforder m on F with respect to D + C. Now gr
n is a complete series (see
14 D. A. Gudkov
[136*]). Since the order of D + C is equal to 3k + 28, we see thatη = m2 ~(3k + 25). By Plucker's formula ([136*]) the class m of Fis equal to m(m - 1) - (3k + 28). Consequently, η = m + m' > 0.Furthermore, the genus ρ of F is equal to \(m - 1) (m - 2) - (k + δ).Hence the order of the canonical series g£~K on F is equal to2p - 2 = m2 - 3m - (2k + 2δ). Since F does not decompose, (4) yieldsk < 3m - 1, hence « = m2 - (2k + 28) - A; > m 2 - 3m - (2£ + 2δ) + 1and η > 2p - 2. Consequently, gr
n is a non-special·series,1 and by theRiemann-Roch theorem we have r = η — p, that is,r - \m(m + 3) - (2/: + δ) - 1. Finally, L contains a unique curve having
F as a component (F itself). Therefore, r' = r + 1, that is,
r - Ν - (2k + δ). This means that (6) is linearly independent. It is noweasy to complete the proof of the generalized Brusotti's theorem for thecase when F does not decompose (and all singular points are real).
d) The case when F decomposes can be reduced to the case just dis-cussed, by induction on the number of components h. Here the use ofMax Noether's theorem is essential.
REMARK. For the case when F has only simple double points (a simplecurve) and does not decompose, Bertini [138*], using the properties of thecanonical series on F, has proved that (6) is linearly independent. Brusotti[32] applied this fact in using the simplification of the singular points ofan indecomposable simple curve. The idea of applying Noether's theoremto the case when a simple curve decomposes is also due to Brusotti.
7. The restriction (4) on the number of cusps in the generalizedBrusotti theorem is not desirable. The removal of this condition isconnected with the problem of the special character of a linear series,which is as follows (see [139*], [140*]).
Let F be an indecomposable curve of order m and genus ρ and gr
n acomplete linear series on F. It is easy to see that if η > 2p - 2, then gr
n
is non-special; if η < ρ - 1, it is special. The question of the specialcharacter of gr
n remains unsolved if ρ — 1 < « < 2 p - 2 .
8. One can deduce interesting consequences from the generalized Brusottitheorem. For example, suppose that Θ is a linearly connected set in RPN
(in the space of curves of order m) and that each Φ Ε Θ has only simpledouble points and cusps, where the number of real simple double points isconstant and equal to δ ' and the number of real cusps is constant andequal to k'. In addition, for each indecomposable component Φ; (of orderm,·) of Φ the number kj of real cusps satisfies the condition
k'j < 3m;- — 1. Then all the Φ G Θ have the same type of degree of non-
roughness s = δ ' + 2k' in RP 2 [116].1 The definition of special and non-special linear series can be found in [ 136* ].
The topology of real project ive algebraic varieties 15
9. We consider another question on the variability of the singular pointsof a simple curve. In the study of algebraic curves it is sometimes necessaryto change, as little as desired, but in other respects arbitrarily, the positionsof the singular points of a curve so that the type of curve in R P 2 doesnot change. In this context we give the following definition (see [112],[113], [116]).
Suppose that F G RPN is a δ-simple curve1 and that z-, 1 < / < δ, areall the singular points of F. We say that k real singular pointsZj, 1 < / < k, of F are variable if for any sufficiently small ε0 > 0 andω > 0 one can find a ν(ε0, ω) "> 0 , such that, having chosen the pointsZj £ Ω (z;-, v), 1 < / " < £ , arbitrarily, there is a continuous curveF Φ C U(F, ε0), such that each Η £ F Φ is a δ-simple curve and itssingular points z- £ Ω*(ζ.·, ω) (1 < / < δ) (of course, for real singularpoints z'j € Ω(ζ;·, ω)), where zj = z;-, 1 < / < k, for Φ. Note that all theΗ € F Φ have the same type in RP 2 . The following facts hold [116]:
NECESSARY CONDITION FOR VAR IABI LITY. For k real singularpoints of a δ-simple curve F Ε RPN to be variable it is necessary that2k + δ < Ν = ψ (m + 3).
CRITERION OF VARIABILITY FOR A RATIONAL CURVE. Suppose
that a curve F £ RPN of order m does not decompose, that it is simple,and that its genus is ρ = 0. Then for any of its k real singular points tobe variable it is necessary and sufficient that
(7) 2k < 3m - 1.
SUFFICIENT CONDITION FOR VARIABILITY. Suppose that F € RPN
does not decompose, that it is a δ-simple curve, and that its genus isρ > 1. Then for k real singular points of F to be variable it is sufficientthat (7) holds.
These facts are proved by the same method as Brusotti's theorem, thatis, varieties of curves in R P ^ are examined, and the Riemann-Rochtheorem is used to establish that certain systems of linear equations areindependent [111], [113],
10. To conclude this section I describe the so-called "marking method"(see [44]). This is often called the "small variation method" or the "smallparameter method".
THE MARKING METHOD. Let Cm and Ck be real non-singular curvesof orders m and k. Suppose that Cm and Ck intersect in m'k distinctpoints, I of which are real: Zj, 1 < / < /. In addition, let dm+k be a realcurve of order m + k intersecting Cm Ck in (m + k)2 distinct points, r ofwhich are real: Ab 1 < i < r. Finally, suppose that the three curves Cm,Ck, and Bm+k do not all have common points. Then for sufficiently small
1 F is called a δ-simple curve if it has S simple double points and no other singularities.
16 D. A. Gudkov
t > 0 the curve
-j- — U
is non-singular, lies (in RP 2) in a small neighbourhood of Cm Ck, and inter-sects Cm Ck in RP 2 only at At 1 < i < r, without tangency. The realbranches of Cm+lc can be constructed by the following method:
1°. We construct Cm and Ck, whose real intersection points Zj are nodesof Cm Ck.
2°. We draw 6m+k in neighbourhoods of the intersection points At of dm+k
and Cm Ck (Fig. 4).3° Near each arc of Cm Ck bounded by two points Ai we place the signs
of the polynomials Cm Ck and 6m+k on each side of this arc.4°. We take any point a of CmCk different from Aj and z-, and we take
a point a in a sufficiently small regular neighbourhood of a of CmCkdm+k
in a domain where CmCk(a) and em+k(a) have different signs.5°. A complete real branch <x of Cm+k passes through a (see the broken
curve in Fig. 4). α runs along the arc of CmCk not intersecting it beforethe nearest (relative to CmCk) point At at which a passes to the other sideof CmCk (without tangency), etc.
6°. The other complete real branches β, γ, . . . of Cm+k are constructedanalogously.
For t < 0 and sufficiently small | t \ the construction of Cm+k is the sameas for t > 0, but in a domain with signs identical to those of CmCk and
Fig. 4
§3. Harnack's theorem and the construction of Λί-curves
l . I n 1876 Harnack [7] proved the following proposition:HARNACK'S THEOREM. Let Cm be an indecomposable algebraic curve
with real coefficients in RP 2 . Let m and ρ be the order and genus of Cm,
The topology of real projective algebraic varieties 17
respectively. Then the number of complete real branches of Cm in RP2
does not exceed ρ + \. In particular, if Cm is a non-singular curve, thenthe number of its complete real branches does not exceedi (m - 1) (m - 2) + 1.
PROOF FOR A NON-SINGULAR CURVE (according to Harnack). It iswell known that for even m a non-singular curve Cm consists only ofovals in RP 2 and that for odd m it consists of one complete odd realbranch and ovals. We assume that Cm of odd order m > A has not lessthan Μ + 1 ovals. On each of the Μ ovals we take one point zt, and onone of the remaining ovals we take m - 3 points yj. Then there exists acurve Cm_2 (of order m - 2) passing through all the zt and y.-. This curvehas more than m(m - 2) common points with Cm. The resulting contra-diction proves the theorem for even m. The proof is analogous for odd m.The proof suggested by Harnack is difficult even in the case when Cm hasonly simple double points. This proof has still not been carried throughfor the case when the singularities of Cm are arbitrary.
In 1891-1892, Hurwitz [11] and Klein [14] gave another proof ofHarnack's theorem. We carry it out for the case of a non-singular curveCm. We denote by A the set of real points of Cm and by CA the set ofpoints of Cm in CP2. It is well known that CA is a two-dimensionalorientable variety of genus ρ = \ (m — 1) (m - 2). Suppose that A con-sists of t complete real branches γ 1 ; y2, • • • , It- These branches are pair-wise disjoint, and each of them is diffeomorphic to a circle. We considerthe involution of complex conjugation conj: CP2 -»· CP2. An involution isthen induced on CA, which we also denote by conj: CA -> CA. Now A isthe set of fixed points of this involution. We take an arbitrary point Ρ on7·. Obviously, one can choose an affine coordinate system with centre atΡ so that the equation of 7,· (in a neighbourhood of P) has the formy(x) = a1x + a2x
2 + • . . , where the ak are all real. It is not hard to seethat the points (x, y(x)) and (x, y(x)) on CA for χ complex and | χ \sufficiently small lie on different banks of jj. It follows that 7, (t > 1)does not divide CA. Otherwise the two components into which 7j dividesCA must be carried into each other under conj: CA ->· CA. However, thisis impossible, because at least one of these components contains fixedpoints (7 2 , . . . , 7 f). Similarly j u y2, . . . , 7f_j do not all together divideCA and hence t - 1 < \ (m - 1) (m - 2), that is,
t < j (m - 1) (m - 2) + 1. Harnack's theorem for a non-singular curve isnow proved.
It follows that for t = \ (m - 1) (m - 2) + 1 the curve A divides CAinto two components that are carried into each other under conj: CA^-CA.For 7, must intersect the last handle of CA, or else j t alone would divideCA or intersect the remaining branches, which is impossible.
Harnack's theorem was proved by Hurwitz [11] and Klein [14] for acurve with arbitrary singularities (see [141*]). Here the Riemann surface
18 D. A. Gudkov
for the given curve is used instead of C4.2. Also in [7] Harnack proved that his estimate is exact. In particular,
for each m he constructed a non-singular curve of order m consisting ofΜ real branches. Petrovskii [56] called such curves Μ -curves.
Fig. 5
HARNACK'S METHOD OF CONSTRUCTING M-CURVES. We take aline Ei and an ellipse C2 that intersects Ex in two points zx and z 2 (seeFig. 5.1; the outer circle is the line "at infinity" of the projective disk.The same applies to the other drawings). Let θ3 = Lx -L2-L3, where L, isa line that intersects Ex at At; Au A2, and A3 are distinct and lie in theinterval zxz2 outside C2. We arrange the signs of the polynomials EXC2
and 0 3 (see Fig.5.1). We construct a curve C3 = EXC2 + ίθ3 according tothe marking method (t > 0, the broken curve). C 3 has an odd branch andone oval, that is, it is an Λί-curve, obviously, of the only possible type1
for m = 3.We take the curve EXC3 (Fig. 5.2). Let 0 4
Ξ LlL2L3L4, where L{ is aline that intersects Ex at At in the interval z2z3 (the At are all distinct).Then C 4 = EXC3 + ίθ4, for the appropriate sign of t and sufficiently small| t |, consists of four ovals (the broken curve according to the markingmethod in Fig. 5.2). One of the ovals of the curve, say a, intersects E^ infour points At. We note that the A( are arranged on Ελ and a in theidentical order, that is, one can establish directions on Ex and a such that
1 Two non-singular curves have the same type if they are isotopic in RP2; see § 1.
The topology of real protective algebraic varieties 19
by moving along Ex and a. continuously in these directions, the Ai are en-countered in the identical order. Now C4 is an Af-curve of the only possibletype for m - 4.
The curve C5 = EXC^ + td5 = 0 is constructed analogously (see thebroken curve in Fig. 5.3) and is an M-curve of the only possible type (form = 5). Now C5 consists of an odd branch a that intersects Ex in fivepoints arranged on a. and El in the identical order and six ovals outsideeach other. The position of C s relative to Ε ι is indicated in Fig. 5.4.Assuming that Cm is an M-curve that intersects Ex in m points that are allarranged on one complete real branch a of Cm in the identical order on aand Ex, it can easily be shown that there exists an analogous Af-curve oforder m + 1. Hence Αί-curves of arbitrary order m can be constructed byinduction.
In Harnack's method the construction is unique up to m = 4. Then forodd m the construction from m to m + 1 is possible along two paths. Forexample, for m - 5 the Ai can be chosen in the interval zxzs or in anyinterval 2;z/+1 (it makes no difference in which); see Fig. 5.4. For even mthe construction from m to m + 1 is always unique.
3. We have seen that for m < 5 the only possible types of ΛΖ-curves canbe constructed by Harnack's method. The situation is entirely for m > 6.Since the sextic curves are the simplest of these, we list the logicallypossible types of ^/-curves C 6.
If C6 consists of k + I ovals lying outside each other and one more(principal) oval enveloping k of these ovals and leaving the other / outside,we say that C6 has type | /. If C 6 consists of k ovals outside each other,we say that C6 has type k. Now C6 consists of 11 ovals, hence the follow-ing types of such curves are possible:
Table 2
3, -f4, ~5, -f 6, ~1, -f 8, -f9,
Only one of these curves can be constructed by Harnack's method,namely, -γ 9 (see Fig. 5.5). A curve is obtained at the next stage (after
C s ) by Harnack's method or by a suitable simplification of the nodes ofE1C5 in Fig. 5.4, using the fact that these simplifications, according toBrusotti's theorem, are independent.
§4. The methods of Hubert and Brusotti for the construction of Af-curves
1. In 1891, Hilbert [10] first raised the question of the mutual dispositionof the ovals of a non-singular plane curve, especially an M-curve. We have
20 D. A. Gudkov
already seen that Harnack's method of constructing M-curves does notdecide the question of the mutual disposition of the ovals for sextic curves.Therefore Hubert sought another method of constructing Λί-curves.
HILBERT'S METHOD FOR THE CONSTRUCTION OF M-CURVES.With this method, the construction is performed separately for even andodd m.
We take two ellipses, say E2 and C2, that intersect in four real pointsz1, ζ2, z3, and z4, arranged on E2 and C2 in the identical order (Fig. 6.1).We form 04 = Lx •L2'L-i'L$ = 0, where L,· is a line intersecting the arcz 3 z 4 of E2 in two points A2i_i and A2i (i - 1, 2, 3, 4). ThenC4 = E2C2 + ίθΛ = 0, for the appropriate sign of t and sufficiently small\t |, consists of four ovals, among which a intersects E2 in eight points A}-arranged on α and E2 in the identical order. Now C4 can be constructedby the marking method (see the broken curve in Fig. 6.1). The dispositionof C 4 relative to E2 is indicated in Fig. 6.2. By Brusotti's theorem we canobtain a curve of type % 1 from E2C4 by a simplification of nodes. If atthe relevant stage of the construction we choose the Aj on the arc zYz^of Ε2, we obtain a curve C4 disposed relative to E2 as indicated in Fig. 6.3.A curve C6 of type j- 9 can then be obtained from E2Cn.
The subsequent construction of Λί-curves for even m is performedaccording to the formula
(8) um+2
where 6m+2= [[ Lt and the m + 2 lines Lt, 1 < / < m + 2, intersect E2i=\
in 2(m + 2) distinct points situated on one of the arcs of E2 into whichCm divides it.
The construction for odd m begins with E2CU where C t intersects E2
in two real points, and also proceeds according to (8).Hubert [10] introduced the concept of a "nest of weight s". A non-
singular curve Cm is said to have a nest of weight s if there exist s ovals
The topology of real projective algebraic varieties 21
for Cm enclosed successively inside each other. The following theorem isproved in [10]:
HILBERT'S MAXIMAL NEST THEOREM. An M-curve of order m cannothave a nest of weight greater than [^ - 1 ] , and M-curves of this weightexist for each m.
The first assertion follows easily from examining the intersection of thecurve with a straight line. The second assertion is proved by constructingthe appropriate Λί-curves by Hilbert's method.
2. In 1910-1917, Brusotti [22], [27], [28], [30], [31] gave the generali-zation described below of the methods of Harnack and Hubert for the con-struction of M-curves. In Coolidge's monograph [46] there are some errorsin this context.
BRUSOTTI'S METHODS FOR THE CONSTRUCTION OF M-CURVES.A simple arc a of a curve En of order η is called a base if it has η · νdistinct points of intersection with a fixed curve of order v. The numberν is called the rank of a.
An M-curve En is called a generating curve if it has two bases a and bwithout common points, where the rank ν χ of a divides η and the rankv2 of b divides In.
An M-curve Cr is called auxiliary to a generating curve En if the follow-ing conditions are fulfilled:
1) All the η · r points of intersection of En and Cn are distinct and lieon one complete real branch e of En and on one real branch c of Cr inthe identical order.
2) If a base a of En lies on e, then a lies entirely on one arc of e arisingfrom the division by a branch c; if e is an odd branch and r is even, thena belongs to the same arc of e that, cogether with one arc of c, forms anodd cycle (that is, intersects a line in general position in an odd numberof real points).
3) The rank vx of a divides η and r.4) The rank v2 of b divides 2n and r.A generating curve En is said to be complete if it has auxiliary curves
Cr of all orders r < n.Let En be a generating curve and Cr auxiliary to En. From the definition
of a it follows easily that there exists a curve Θη+Γ (equal to the product of(n + r)lvx close curves of order vx) that intersects the base a of En inn(n + r) distinct points. Then
(9) Cn+r =En-Cr + t0t+r = 0
for sufficiently small | t | and the appropriate sign of t is an M-curveauxiliary to En. The process of construction can be continued with b, andat the next stage we return to a, etc. Thus, a sequence of M-curves isconstructed of orders m - k η + r (k = 1, 2, . . . ).
22 D. A. Gudkov
In Harnack's method the line Ex is the generating curve with C1 as anauxiliary line. In Hubert's method the generating curve E2 has the auxiliarycurves C t and C2.
An arbitrary simple arc a of Cn (for η > 3) cannot be a base (the num-ber of linear conditions is too large). Brusotti [28], [31] proved thefollowing facts:
1°. There exist complete generators E3, £ 4 , and E5.2°. There exist generators En for each η > 6 (however, complete gener-
ators have not been found).We note that a curve Cn auxiliary to En can be obtained from En by the
formula
(10) θη = 0,
where θη is a curve of order η intersecting the base a of En in n2 distinctpoints. A sequence of Μ -curves of orders m = n'k (k - 1, 2, . . . ) canbe constructed from En and Cn in accordance with (9). The constructionof Λί-curves in accordance with (9) and (10) for η > 3 is called Brusotti'smethod for n, and the resulting Μ -curves Brusotti's n-th series.
As an example I indicate how Brusotti constructs generating curves(one of the methods).
Fig. 7
Let Li, L2, and Z-3 be three lines that do not all pass through the samepoint (see Fig. 7.1), and let / l 5 l2, and /3 be lines such that the curve$3 = lxl2l3 intersects each Lt in three points not separated by the pointsof intersection of the L,·. Then the curve E3 =LiL2L3 + t63 = 0 forsufficiently small | ί | > 0 and the appropriate sign of t can be constructedby the marking method (the broken curve in Fig. 7.1). Now E3 has an ovaland an odd branch α that intersects each L{ in three points. It is obvious(see Fig. 7.1) that E3 is a generating curve with three bases of rank 1.
We take L 1 ; L2, L3, and E3 (see Fig. 7.2). Next, we take four lines/,· (/" = 1, 2, 3, 4), each intersecting an odd branch α of En in the base a
The topology of real projective algebraic varieties 23
as indicated in Fig. 7.2. Then the curve E4 Ξ E3'L1 + ίθ4 = 0 forsufficiently small | t | and the appropriate sign of / can be constructed bythe marking method (the broken curve in Fig. 7.2). Now E4 is a generatingcurve having three bases of rank 1. It is not hard to show that E3 andE4 are complete generators.
Continuing the construction by induction, we can construct a generatorEn with three bases of rank 1 for arbitrary n. Using the constructions inHubert's method, Brusotti constructs generators En (for arbitrary η > 3)having one base of rank 1 and one base of rank 2. I mention that forsextic curves Brusotti's methods do not yield anything new in comparisonwith the methods of Harnack and Hubert.
3. In 1923 Wiman [35] discovered in the methods of Harnack and Hil-bert new possibilities for the construction of Af-curves with certaincollections of nests.
WIMAN'S METHOD FOR THE CONSTRUCTION OF A/-CURVES.Suppose that Ck is an Af-curve obtained by Harnack's method (with genera-tor Εχ). Then some branch a (of Ck) intersects Ex in k points, that is, thereis a base a of rank 1 on ex.. Therefore, there exists a curveQk = L1 'L2 · . . . 'Lk, where Llf L2, . . . , Lk are lines close to E1; thatintersects a in k2 distinct points on a. Now Ck = Ck + tdk = 0, forsufficiently small | / | > 0, is an M-curve, and the branch α of Ck intersectsa in k2 distinct points arranged on a and a in the identical order. Henceit is not hard to see that C2/t = Ck-Ck + r\p2k =0, where \p2k
n a s n o r e a l
points, | r | > 0 is sufficiently small, and τ has the appropriate sign, is anM-curve of order 2k. This construction, however, cannot be continued. ForCk is not, in general, a generating curve, because it has only one base.Wiman used the curves E2'Ck obtained in Hubert's method (for even k)in a similar way.
We say that two nests of Cm lie outside each other if the outermostovals of these nests lie outside each other. In the paper in question Wimanproves the following proposition:
WIMAN'S THEOREM. For each even order in = 2k there exists anM-curve Cm having \ (k - l)(/c - 2) nests of weight 2 lying outside eachother {one oval lies in each outer oval of the nest) and k2 ovals lying out-side each other and outside the nests of Cm .
To prove this it is sufficient in passing from Ck to Q + 1 in Harnack'smethod to take the points Aj (see above) on the boundary of the ovaloidformed by Ex and the branch of Ck in which there are ovals of Ck(k^5).Wiman's method must be applied to the curve Ck thus obtained.
4. Two articles by Biggiogero [33], [34] in 1922-1923 also deal withthe construction of A/-curves with certain collections of nests. Here Brusotti'smethods are used to construct Af-curves. Biggiogero introduces the conceptof a maximal inclusion. Suppose that a non-singular curve Cm of order m
24 Λ /4. Gudkov
has two ovals, α and β, inside of which there are no ovals of Cm . Let Ρand Q be two points inside a and β, respectively. If Cm hasr > 1 (s > 1) ovals enclosing Ρ (Q), but not Q (P) and ? ovals enclosingboth Ρ and Q, then it is not hard to see (by the intersections with the linePQ) that r + s + / < [m/2]. We say that Cm has the maximal inclusion{r, s) if r + s + t = [mil]. We may assume that r > s (and this is donein what follows). The following facts are proved in [33], [34] by themethods of Hubert and Brusotti:
BIGGIOGERO'S THEOREM. There exist M-curves Cm of the followingorders having the indicated maximal inclusions:
No. Order Maximal inclusion
1
2°
3°
4°
5°
6°
m
m
mm
m
m
> 6= p·*
= Ah
= 4fc
= pft Η
= Ah
ι, ρ even,
+ 2
- 1, ρ even
+ 1
ρ > 6
, p > 6
(r, 1) where 2 < r < \
(hy, h\ where 2 < y <
(h,h)(h + 1, h)
{hy, h), where 2 < y <
(h,h)
We note that the inclusion ([(m—2)/2], 1) constructed in Hubert's maxi-mal nest theorem is the special case 1° of Biggiogero's theorem. We have tokeep in mind that the concept of maximal inclusion introduced byBiggiogero deals only with the restriction imposed by the intersection witha straight line. For example, according to 3° of Biggiogero's theorem, anoctic M-curve may have the inclusion (2, 2), that is, two nests of weight 2.However, by Wiman's theorem there exists a curve C8 having three nestsof weight 2.
In 1939 Farina [60] supplemented Biggiogero's results by proving thefollowing proposition with Brusotti's method:
7°. There exists an M-curve Cm of order m = Ah + 3 having the maximalinclusion (k + 1, k).
Other interpretations, refinements, and supplements of the works ofHarnack, Hubert, and Brusotti we have examined are given in [12], [16],[29], [36], [51], [52].
§5. .M-curves on a quadric
It was natural to attempt to extend the results obtained by Harnack forplane curves to space curves. Hubert studied this problem in 1891 in thepaper [10] quoted above. I recall that the order η of an algebraic spacecurve Cn is defined to be the number of intersections (counting imaginary
The topology of real projective algebraic varieties 25
points and multiplicities) of the curve with a plane (not containing infinitelymany points of the curve). Hubert proved the following analogue to Harnack'stheorem.
THE HARNACK-HILBERT THEOREM. Let Η = 0 be the equation of anon-singular real quadric surface in R P 3 . Let Cn be an irreduci-ble algebraic curve of order η lying on H. Then the number of completereal branches of Cn does not exceed \{n—2)2 + 1 for even η and\{n-\) («-3) + 1 for odd n.
PROOF. Suppose that the surface Η = 0 is a hyperboloid of one sheet.We take a real point Ρ on Η and suppose that Ρ does not lie on Cn. Letω be a plane that intersects Η in an ellipse and does not contain P. Theneach linear generator of Η passing through Ρ intersects ω in one point. Wedenote these points by A and B. We project Cn from Ρ onto ω. In pro-jecting we obtain a curve Cn. It is obvious that Cn and Cn have the sameorder and the same number of complete real branches. In addition, thesecurves have the same genus, because in this case the projection is a birationaltransformation. The tangent plane to Η at Ρ intersects Cn in precisely ηpoints (which may be assumed to be distinct), hence Cn has singular points(with distinct tangents) at A and Β of multiplicities nl and n2, wherenx + n2 - n.
If the plane curve Cn of order η does not decompose and has singularpoints at A and Β of multiplicities nx and n2, where nx + n2 - n, thenthe genus of Cn does not exceed \{n-2)2 for even η and \{n-\) (n~3) forodd n. It is not hard to see that for even η the genus of Cn is equal to\{n-2)2 if «! = n2 - n/2, the tangents to Cn at A and Β are distinct, andCn has no other singular points. Similarly for odd η the genus of Cn isequal to \(n~\) (w—2) if nx - \{n + 1), n2 = \{n — 1), the tangents to Cn
at A and Β are distinct, and Cn has no other singular points. By Harnack'stheorem the number of complete real branches of Cn (and hence of Cn)does not exceed \{n~2)2 + 1 for even η and \{n — \) (n~3) + 1 for odd n.
If Η is an ellipsoid, then A and Β are imaginary, but the proof isunchanged. The theorem is now proved.
2. Next, Hubert proved by an inductive construction that there existcurves Cn on a quadric Η with the maximal number of branches indicatedin the last theorem,1 which we call Μ -curves.
HILBERT'S METHOD FOR THE CONSTRUCTION OF Af-CURVES ONA QUADRIC IN R P 3 . We carry out the construction for a hyperboloid ofone sheet H. We intersect Η with the plane Ε - 0 in an ellipse g. Nextwe take a plane Fx - 0 that intersects Η in a curve C2 and "6 in twopoints Α γ and A2. On one of the arcs AXA2 of i we choose four points
A complete real branch of a space curve is said to be even (odd) if a real plane in general positionintersects it in an even (odd) number of real points. In this section instead of "complete real branch"of a curve we say briefly "branch". An even branch without singular points is called an oval.
26 D. A. Gudkov
Bh 1 < i < 4, and through the pairs BXB2 and B3B4 we pass planes Lx
and L2, respectively, different from E. Then the surface
(11) F2 = EFl + tLxL2 = 0
for sufficiently small | t | > 0 and the appropriate sign of / intersects Η inan Λί-curve C 4 consisting of two branches (see the broken curve in Fig. 8.1),one of which, say a, intersects % in the Bi arranged on g and ex. in thesame order.
Fig. 8
We take the curve C4 of intersection of Η = 0 and F2 = 0. We denotethe points of intersection of C4 with % by Ax, . . . , A4; on one of thearcs 4̂,v4,+i we choose six points Bj\ and through the three pairs of thesepoints we pass planes Lx, L2, and L3, different from E. Then the surface
(12) F3 Ξ EF2 + tLxL2L3 = 0
for sufficiently small \t \ > 0 and the appropriate sign of t intersectsΗ = 0 in an Λί-curve C6 consisting of five ovals, one of which, say a,intersects § in the B.- arranged on if and a. in the identical order. Further-more, it is easy to show by induction that there exist Λί-curves of arbitraryeven order n.
3. To obtain Af-curves of odd order η the construction is modified some-what. At the first stage the plane Fx = 0 is taken for the tangent to Η atsome point Ρ not lying on g. Now Fx = 0 intersects Η = 0 in two lines:Cj, a curve of the first order, and /, a fixed line intersecting % at A (seeFig. 9.1). On an arc of % (abutting A) we choose three points Bx, B2,and B3 (in the order A, Bx, B2, B3); through / and Bx we pass a planeLi, and through B2 and B3 we pass a plane L2, different from E. Thenfor sufficiently small \t \ > 0 and the appropriate sign of t, (11) intersectsΗ = 0 in / and in an Λί-curve C 3 consisting of the one odd branch αindicated in Fig. 9.1 by the broken curve. This branch intersects % in threepoints arranged on % and ot in the identical order.
We take Η and F2, which intersect in / and C 3, and we rename the points
The topology of real protective algebraic varieties 27
Fig. 9
of intersection of C 3 with % by Αλ, Α2, and A3 (see Fig. 9.2). These pointsdivide Ε into arcs. On the arc containing A we take five points Bf- arrangedin the order.Α, Βγ, B2, . . . , B5. We pass a plane Li through / and Bu
and through each of the pairs B2, B3 and 5 4 , B5 we pass planes L2 andL3, different from E. Then for sufficiently small | t \ > 0 and theappropriate sign of t, (12) intersects Η in an Af-curve C s consisting of threebranches. One of these branches, say a, intersects % in five points lying on% and α in the identical order. The rest of the construction by inductionis obvious.
If the quadric Η - 0 is an ellipsoid, then only Af-surfaces of even ordersη can be constructed on Η by Hubert's method, and all their branches areeven.
4. Hubert also investigated the question of how many odd branches anΛΖ-curve situated on a hyperboloid can have, and he proved the followingproposition:
THEOREM ON THE NUMBER OF ODD BRANCHES OF AN A/-CURVEON A HYPERBOLOID. Suppose that an M-curve Cn of order η situatedon the hyperboloid of one sheet Η = 0 has ν odd branches.
< 2k - 2;< 2.= 0.< 2/c - 1;< 3.< 2k - 1;
and η - 3, then ν = 1.
M-curves with the greatest number of odd branches indicated in l°-4°exist.
5. The study of curves on a quadric especially A/-curves, by the "smallparameter" method was continued in [26], [57], [61], [63], [68], [77],[97].
1 .
2°.3°.
4°
If η = 4k
If η = 4k ΗIf η = 4k Η
If η = 4k Η
^ 2,h 1
Η 3
andand
andandand
η > 4,η = 4,
η > 5,η = 5,η > 3,
thenthenthenthenthenthen
V
V
V
V
V
V
28 D. A. Gudkov
Curves lying on a non-singular cubic surface, on a cubic cone, and oncertain other surfaces of a particular form in R P 3 have also been investi-gated. I confine myself to pointing out the literature: [ 3 9 ] - [ 4 2 ] , [ 4 5 ] ,[ 4 9 ] , [ 6 4 ] , [ 6 6 ] , [ 6 9 ] , [ 7 0 ] , [ 7 3 ] , [ 7 5 ] , [ 8 4 ] , [ 8 5 ] , [ 1 0 6 ] .
§6. The theorems of Petrovskii and Oleinik
1. In 1891, Hubert [10] made a conjecture about sextic curves: a sexticΛί-curve cannot have type 11 (eleven ovals outside each other). He said thathe arrived at this assertion in an "extraordinarily complicated way".
2. In 1906, V. Ragsdale [17] set herself the task of finding an experi-mental law generalizing Hubert's conjecture on sextic curves to curves ofeven orders m > 6. The experimental material were the curves constructedby the methods of Harnack and Hubert (see § § 3 and 4). She called anoval of a non-singular curve / "internal" if it lies inside an odd number ofother ovals of /, and "external" otherwise. Later, Petrovskii [79] calledsuch ovals odd and even, respectively. We shall use these more felicitousterms.1 We denote the number of even ovals of a non-singular curve / by
Ρ and the number of odd ovals by L. The following results are containedin [17]:
1°. It is proved that for an M-curve of even order m constructed byHarnack's method always Ρ = 3m*-6m + \ and L = (ΟΙτΙΆΈζ*!.
ο 8
2°. It is proved that for an M-curve of even order m constructed byHilbert's method the following inequalities hold:
(14)
3°. For each even m, an M-curve with Ρ = 3«!z«* + 1 and L = (m-2m-^_8 8
is constructed by Hilbert's method.4°. For each m = 2 mod 4, an M-curve with Ρ = i^2Km=l)_+ 1 and
L = ^nl^HL is constructed by Hilbert's method.8
5°. The following conjecture is made:RAGSDALE'S CONJECTURE. For a non-singular curve of even order m
in RP2, the following estimates hold:
(15) / '< : W 7 6 ' * + 1 ,
(1(3) L ^ 3 w 2 7 6 m .
Even they aie also not quite successful because the terms "even and odd (complete real) branch" havea different meaning. I cannot suggest better terms, and I hope that no confusion results.
The topology of real protective algebraic varieties 29
If these estimates are correct, then the examples of Ragsdale's construction(see above) show that (15) is sharp for any even m and that (16) for anym = 2 mod 4.
6°. Ragsdale noted that for a non-singular curve of even order m, (15)implies the weaker inequality Ρ - L < 3m1-6m + ^ which perhaps is easierto prove than (15).
I mention that since Ρ > 1, it obviously follows from (16) that
L - Ρ < 3 m ' 6 m - 1. Both these inequalities can be conveniently combined8
in the following proposition:RAGSDALE'S WEAK CONJECTURE. For a non-singular curve of even
order m in RP2, the following estimate holds:(17) \P-{L + l ) \ ^ ^ ^ - .
Ragsdale, using a slight modification of Harnack's method, constructed anon-singular curve of arbitrary even order m for which Ρ = 3m2-6m + 1
8
and L = 0. For such a curve (17) holds with the equality sign.7°. Finally, she mentions that for an Λί-curve of even order m her con-
jecture (for Ρ > L) means that Ρ - L < (m/2)2. I add that for L > Ρ herconjecture for M-curves means that L — Ρ < (m/2)2 — 2. Therefore, for anM-curve of even order m Ragsdale's conjecture is equivalent to the estimate
(18) | P- (L + 1)| < (m/2)2 - 1.
Ragsdale's conjecture (see 5°) has still not been proved or disproved.3. In the period from 1891 to 1933 many mathematicians attempted to
prove Hubert's conjecture on sextic curves; however, until Petrovskii'spapers [50] and [56] these attempts were unsuccessful. Petrovskii in [56]gave remarkable proofs of two general theorems on the mutual dispositionof the ovals of non-singular plane algebraic curves. The first of thesetheorems can be conveniently stated in the following form:
PETROVSKII'S FIRST THEOREM. Fora non-singular curve f of evenorder m in RP2, the following estimate holds:
(19) | 2 ( / > - L ) - l | < 3 < w 2 - 6 w -Li.
I mention that from Petrovskii's first theorem it follows that Hubert'sconjecture on sextic curves is true, because for m = 6 (19) yieldsI P - L | < 10. Petrovskii proved that (19) is sharp, by constructing a curve ofarbitrary even order m for which Ρ - 3 w 2 ~ 6 m + 1 and L = 0, by a modificationof Harnack's method. He discusses where (19) is sharp for L > P. In [56] ,1
for each m = 2 mod 4 he constructs by a modification of Hubert's method
Petrovskii did not know of the existence of Ragsdale's paper [17]. In [56], therefore, he did not note thathis first theorem had been stated by Ragsdale as a conjecture. I mention that Petrovskii repeated tworesults of [17]: 1) he constructed a curve of even order m for which Ρ = j (3m2 -6m) + 1 and/, = 0;2) he conjectured that for a non-singular curve of even order m both Ρ and L do not exceedi(3m2-6m) + 1.
30 D. A. Gudkov
a non-singular curve of order m for which Ρ - 1 and L = 3 m *~ 6 w . Further-more, for each m = 0 mod 4 he constructs (by the same method) a non-singular curve of order m for which Ρ = 1 and L = 3 m ' ~ 6 m - 2.
I give a brief account of a proof of Petrovskii's first theorem.
1) We denote the coordinates of a point in CP2 by [x 0: x\'- X2^- Alongwith CP2 we consider the affine plane C2 (x0 = 1) in which the equationof the curve f(x0, x 1 ; x 2 ) = 0 has the form f{\, x, y) = F(x, y) = 0. Irecall that the solutions (xy, y}-) of the system of equations
(20) Fx(x, y) = 0 , Fy = (x, y) = 0
are called the critical points of F.2) It is not hard to show that by an arbitrarily small change of the
coefficients of f(x0, Χι, x2) (with its order and topological disposition inRP2 preserved) we can achieve that: a) all (m — I ) 2 critical points arefinite (that is, lie in C 2 ) and pairwise distinct; b) the curve / = 0 hasprecisely m distinct points of intersection with x 0 = 0. Suppose that k ofthese points are real; c) the value of F at the real critical points (the realcritical values) are pairwise distinct. Henceforth we assume that a), b), andc) hold. We also assume that the coordinate system in R P 2 is chosen sothat k > 0.
3) Petrovskii calls an oval α of a non-singular curve φ(χ0, xi; x 2 ) ofeven order m positive if the polynomial φ changes sign from positive tonegative as [x0: x t : x 2 ] passes from the interior of a to the exterior.Otherwise a is said to be negative. We denote the number of positive ovateof φ by ρ and the number of negative ovals by /. If the even ovals for
/ = 0 are positive, then ρ = Ρ and I = L for /. We assume that this condi-tion holds, by changing the sign of / if necessary.
4) We introduce the notation
(21) J(x, y)=yx ryy
Let (Xj, yj) be a real critical point of F. If J(xj, j>;·) > 0 (< 0), then(x-, y-) is said to be positive (negative).
5) We consider the curves F(x, y) - c = 0, where c decreases from avalue c 1 ; greater than all the real critical values of F, to a value c2, lessthan all these values. We watch for a change in ρ - I of the curveF - c = 0. Suppose that (x ;, yf-) is a real critical point and that there areno critical points of F in the interval [F(x;-, yj) + ε, F{x},yj) — ε] exceptF(Xj, y^. Let c decrease from F(x;·, y^) + ε to F{xh ijj) — ε. 1° IfJ(Xj, yj) > 0 then ρ - I for F - c = 0 increases by 1; 2° ifJ(Xj, yj) < 0 then ρ - I always decreases by 1, except only when ρ - Idecreases by 2. In this exceptional case F(x;·, y;·) < 0 (by the choice ofsign for f).
The topology of real profective algebraic varieties 31
6) The curve F(x, y) - cx = 0 has k/2 positive and 0 negative ovals;F(x, y) — c2 = 0 has 0 positive and k/2 negative ovals. Therefore, it followsfrom 5) that the difference between the number of negative and positivecritical points is k — 1.
7) It is not hard to show that if Fx(x, y) and F2(x, y) are two curvesof order η intersecting in n2 distinct finite points and if h(x, y) is a poly-nomial of degree / < n, h ψ 0, then h(x, y) cannot vanish at more thanI·n points of intersection of Fx and F2.
8) Let Γ be a fixed non-zero complex number and h{x, y) a polynomialin χ and y with real coefficients. Then, the' condition thatRe {T-h2(x0, y0)} = 0 at some point (x0, y0) can be ensured by a homo-geneous linear equation with real coefficients with respect to thecoefficients of h(x, y).
PROOF. Let Τ = a + ib and h(x0, y0) = U + iV. Here a and b are realand b Φ 0; U and V are real linear forms in the coefficients of h. We havette{T-h2{x0, ?/„)} = aU2— aV2 — 2bUV. If a = 0, then the equality—2b UV - 0 can be ensured by one of the equalities U - 0 or V - 0. Ifα Φ 0, then the polynomial aU2—aV2—2bU'V has real roots 5χ and s2
with respect to (U: V), hence Re {T -h2(x0, y0)} = a(U — SiV)(U —~s2V),that is, the requisite equality can be ensured by any one of the equalitiesU - sx V = 0 or U - s2 V = 0.
9) Since the critical points of' F are all finite and pairwise distinct, bythe Euler-Jacobi theorem [144*] we have
{Xj yj) η= 0 '
where / is the Jacobian (21); P(x, y) is an arbitrary polynomial of degreelower than (2m — 4), and the sum is taken over all the critical points ofF.
10) I n ( 2 2 ) w e c h o o s e P(x, y) = Fx(x, y)[h{x, y)]2, w h e r e h(x, y) is a narbitrary polynomial of degree \(m — 4) and
(23) Fx{x, y) = mF(x, y) - xFx{x, y) - yFvi$, y).
The degree of Fx is not higher than m — 1. It is very important for whatfollows that F and Fx have the same sign at each real critical point,because Fx(Xj, yf) = m · F(Xj, yf).
11) We note that F — cx = 0 has k/2 positive and no negative ovals andchat F = 0 has Ρ positive and L negative ovals (see above). Therefore, asc decreases from cx to 0 the curve F - c = 0 gains Ρ + a — k/2 positiveovals and L + β negative ovals and loses a positive ovals and β negativeovals.
Hence, by 5), the number of positive real critical points for whichFix,-, yj) > 0 is equal to Ρ + a + β - k/2. We denote this number by A.
32 D. A. Gudkov
Also the number of real negative (/ < 0) critical points for whichF(Xj, y,-) > 0 is equal to L + a + β. We denote this number by Β (theexceptional case holds when F < 0; see 5)).
We denote by B' the number of positive critical points for F < 0 andby A' the number of negative critical points for F < 0. Finally, we denoteby 2γ the number of complex critical points. Then, by 6), we haveΒ + A' - (A + B') = k - \. In addition, it is obvious thatA + A' + Β + B' + 2y = (m - I ) 2 . Hence we easily find that
It is important that F/J > 0 at A + A' critical points and that F/J < 0 atΒ + B' critical points.
12) We assume that
(25) A + A'+y^m(m-2) — i.
We require that h(x, y) (see 9)) vanishes at all A + A' points (whereF/J > 0) and that the real parts of the summands P/J for the complexcritical points in (22) vanish. According to 8), the latter can be ensured by7 homogeneous linear equations with real coefficients with respect to thecoefficients of h. Therefore, there are altogether A + A' + y such equations.Consequently, there exists a non-zero solution of these equations (thedimension of the space for h is equal to \m(m-2)-\). Let h°(x, y) bethe polynomial corresponding to this solution. For this polynomial the sumof all the terms in (22) involving the complex critical points vanishes,because the complex points occur in conjugate pairs, and Ρ and / are real.Finally, we recall that according to 7), h°(x, y) does not vanish at atleast one of the critical points where F/J < 0. For \m(m-2)-y-\ > 0,because otherwise (25) cannot hold. But then the number of real criticalpoints is {m — I) 2 — 2 y > (3m 2-6m + 8)/4, whereas the number of criticalpoints at which h° = 0 cannot exceed i ( m - 4 ) ( m — 1) according to 7).Therefore, the left-hand side of (22) is negative. The resulting contradictionshows that (25) is false, that is, A + A' + y > | m ( m - 2 ) - 1; similarly,Β + B' + y ~> \m(jn — 2) — 1. Now (19) follows from these inequalities and(24), and the theorem is proved.
For a non-singular curve f(x0, Χι, *Ί) ~ 0 of odd order m in RP 2 it iseasy to construct a curve by Harnack's method consisting of one completeodd branch and \{m— 1 )(m — 2) ovals lying outside each other. Consequently,for a curve of odd order there can be no restrictions on \P — L \ other thanthat imposed by Harnack's theorem. Petrovskii, however, found an interest-ing analogue to his first theorem even for the case of curves of odd order.
Let f(x0, X\, Xi) = 0 be the equation of a real non-singular curve of oddorder m and [x0: xx: x2] the coordinates of a point in CP 2 . Suppose that
The topology of real projective algebraic varieties 33
/ = 0 intersects the line x0 = 0 in m distinct points of which k are real.Suppose that the equation of the curve in the affine plane C 2 (x 0 = 1) hasthe form /(I , x, y) = F(x, y) = 0. Then an oval α lying in R 2 (x 0 — 1) issaid to be positive (negative) if F changes sign from plus to minus (fromminus to plus) as (x, y) passes from the interior of α to the exterior. Wedenote by ρ the number of positive ovals and by / the number of negativeovals. We denote by Δ+ (Δ") the number of positive (negative) infinitedomains of RP2, that is, the domains where F > 0 (F < 0) abutting theline x 0 = 0 in RP2.
PETROVSKII'S SECOND THEOREM. For a non-singular curve of oddorder m in R2 the following sharp estimate holds:
(26)Δ+ — Δ" k — l
2
Petrovskii proved that (26) is sharp by constructing (by a modificationof Harnack's method) for each odd m a non-singular curve of order m forwhich
= m, A+ = m, Δ- = 1, ρ =
The proof of the second theorem is similar to that of the first.I note that (26) imposes restrictions on the disposition of the ovals of a
decomposable curve / ·χ 0 = 0 of even order m + 1. This suggests thequestion whether one can deduce Petrovskii's second theorem from the first.The number Δ+ of positive domains of the curve F = 0 in R2 makes itpossible to obtain by a small correction from / ·χ 0 = 0 a non-singularcurve of even order m + 1 having ρ + Δ+ positive ovals and / negativeovals. Therefore, from (19) (counting the even ovals as positive) we obtain
(27) |ρ_Ζ + Δ Ί < ^ 5 .
It follows from (26) that
(28) l P - Z + A+|
Now (27) and (28) coincide for k = m. However, (28) does not followfrom (27) for k < m and hence (26), that is, Petrovskii's second theorema fortiori does not follow from (27), that is, from Petrovskii's firsttheorem.
It should be emphasized that Petrovskii was the first to obtain in [56]theorems that impose non-trivial restrictions on the mutual disposition ofthe ovals of plane curves. I call restrictions obtained from arguments onthe intersection of a curve with a straight line or a conic trivial.
4. In 1949 Petrovskii and Oleinik [82] generalized Petrovskii's theoremsabove to non-singular hypersurfaces in RPq.
For a non-singular curve / = 0 of even order m we denote by B+ thedomain of points of RP 2 , where / > 0. Then Petrovskii's first theorem canbe written in the following form: the inequality
34 D. A. Gudkov
(29)
where χ(Μ) is the Euler characteristic of M, holds independently of whetheror not B+ is orientable.
In this form Petrovskii's theorems are generalized in [82].Let f(x0, X\, . . . , xq) be a non-singular1 non-homogeneous polynomial
of degree m with real coefficients, and let
(30) /(I , xu x2, . • ., xq) = F (xu x2, • · ·, Xq)·
We denote by A the set defined by / = 0 in RPq, by M+ (Mc) the setdefined by F > 0 (F > c) in R<? (x0 = 1), and by B+ (Bc) the closure ofM+ (Mc) in RPi.
A solution Pa(%ia, • · . ^ a ) , Pa €= CPq, of the system of equations
is called a critical point of F, and F(Pa) is called a critical value of F.It is not hard to prove the following almost obvious facts:a) Under an arbitrary sufficiently small change in the coefficients of /
the topology of A does not change. The coordinate system in RPq can bechosen so that this also holds with respect to M+ and B+.
b) Under an arbitrarily small change in the coefficients of / it can beachieved that (31) has (m - \)q distinct finite (that is, lying in C?, x0 = 1)solutions Pa, and that the critical values of F at the real critical points areall distinct and non-zero.
We assume henceforth that / satisfies b) and that the coordinate systemin RPq is chosen so that the topology of M+ and B+ does not changeunder a sufficiently small change in the coefficients of / .
ALGEBRAIC LEMMA, ΰ ^ an arbitrarily small change in the coefficientsof F it can be achieved that there is a polynomial g(xl, . . . , xq) with realcoefficients satisfying the following conditions:
1°. the degree of g does not exceed I = |gm~~" g ~ m | , where [x] isthe integral part of x.
2°. g is equal to zero at each of s(q, m) — y — \ points arbitrarily chosenamong the real critical points of F. Here 27 is the number of complexcritical points of F, and s(q, m) is equal to the number of terms of thepolynomial q
(32) Π %^~whose degree does not exceed I.
3°. g satisfies the inequalities
(33) R
That is, there does not exist a point [x0: x,: . . . '. xQ\ of CPQ at which ^ί— = 0, 0 <ί ί < q,simultaneously. i
The topology of real projective algebraic varieties 35
for each pair Ρβ, Ρβ of complex conjugate critical points. Here Αβ and Αβ
are non-zero complex conjugates preassigned for each pair of complex con-jugate critical points Ρβ, Ρβ.
4° All the values g(Pa) for real points Pa and all the Re {Α^2(Ρ&)} forcomplex pairs Ρβ, Ρβ do not vanish simultaneously.
I do not present a proof of this lemma (see [82]) owing to lack ofspace. I mention that (33) cannot be replaced by equality, because otherwisetoo many linear equations in the coefficients of g would result. Theconsiderable difficulty in proving the algebraic lemma is related to this.
THE PETROVSKII-OLEINIK LEMMA. Suppose that F(xu . . . , xq) is anon-singular polynomial of degree m whose critical points are all finite anddistinct, and that the critical values are mutually distinct and satisfy thecondition c2 < F(Pa^ ^ ci> w^ere c1 and c2 are fixed real numbers. Thenthe manifolds B+, Bc , and Bc satisfy the inequality
(34)
where χ(Μ) is the Euler characteristic of M.PROOF. 1) We note that the critical values of F are non-zero, because
F is a non-singular polynomial. We reduce c monotonically from cx to 0.Then x(Bc) can vary only as c passes through the critical value C* = F(Pa).In a neighbourhood of Pa the function F is representable in the form
F(Xi, x2, .. ., xq) = Ca + ^ x'i2— 2 χ?-^ ΗΆ (x\, . . ., x'q),i=l i=fc+l
where the x\ are local coordinates in a neighbourhood of Pa and H3 con-tains terms of degree > 3 with respect to x\. This representation ispossible, because the Jacobian of Ft at Pa is non-zero. As c decreases andpasses through Ca, x(Bc) increases by 1 if A; is even and decreases by 1if k is odd (see [142*], [143*]).
Suppose that for the values of c between ct and 0 there are A criticalpoints with even k and Β critical points with odd k, and that for thevalues of c between 0 and c2 there are B' critical points with even k andA' critical points with odd k. Then
(m — I ) 5 — 2y =
X
hence
(35)
Therefore, the quantity F/J has the sign (~\)q at A + A' points Pa andthe sign (~l) 9 + 1 at Β + B' points, where /(x l 5 . . . , xq) is the Jacobian of
36 D. A. Gudkov
Fl, . . . , Fq with respect to xx, . . . , xq.2) We claim that each of the numbers A + A' and Β + Β' is greater
than s{q, m) — y — 1. By the Euler-Jacobi formula (see [144*]), for everypolynomial fl{xi,...,xq)oi degree not exceeding q-(m — 2) — 1
(36) 2 !Hfer °·α
where the sum is taken over all critical points Pa of F. We assume that
A + A' < s(q, m) - γ - 1. Then by the algebraic lemma there exists a
polynomial g with real coefficients satisfying the following conditions:
a) g is equal to zero at the A + A' real critical points where
•(-\)q~< 0; b) for each complex point Pe of F we have
i?B = ReiA^iP^i-i)^1 > 0, where A^ = j ^ ; c) the degree of gdoes not exceed / = [ \{qm — 2q- m)]; d) all the values g(Pa) for the realcritical points Pa and all the quantities R^ for the complex conjugate pairs ofcritical points (Pe, P-s) do not vanish simultaneously.
f p q
In (36) we choose /i = (mF — ^XiFj) -g2. Then the degree of/Ί does noti = l
exceed q(m - 2) - 1, and (36) holds. All the imaginary terms on the left-hand side cancel, because the complex critical points occur in conjugate
pairs, and only the summands T7W\8*(Pa) (with sign (~l)q+1 or zero) fromJ\"a)
the Β + Β' real critical points Pa remain, in addition to the quantitiesRp for the complex conjugate pairs of critical points. Among thesequantities there are non-zero ones, and all these have the sign (— l ) i + 1 .The resulting contradiction proves that A + A' > s(q, m) — γ — 1.Similarly Β + Β' > s(q, m) - y - 1. Therefore, (34) follows from (35),and the lemma is proved.
This lemma is the main result of [82]. The next two theorems followeasily from this.
FIRST PETROVSKII-OLEINIK THEOREM. Let f = 0 be the equationof a non-singular hypersurface of even order m in KPq. Then
(37) \%{B+)\^±.{m-i)*-s(q,m) + ±.
The theorem follows from the Petrovskii-Oleinik lemma and from theequality
(38) χ ( β β 1 ) + χ ( 5 β ϊ ) = ΐ ,
which is proved (for even m) in [82].COROLLARY. Suppose that under the conditions of the first Petrovskii-
Oleinik theorem q is odd. Then
(39) | χ(Α)| < {m - I) 3 - 2s(q, m) + 1.
This follows easily from the first Petrovskii-Oleinik theorem.SECOND PETROVSKII-OLEINIK THEOREM. Let f = 0 be the equation
The topology of real projective algebraic varieties 37
of a non-singular hypersurface of odd order m in RPq. Suppose that theplane x0 = 0 is in general position with the hypersurface / = 0 and that/ ( I , x ! , . . . , χ Ξ ^(xi, . . · ,Xq) - 0 is the equation of the part of thishypersurface in Rq (x0 = 1).
1°. If q is odd, then
(40) \χ(Β, ( r a-l) '- S( ?, m)
2°. If q is even, then
(41) -*.(m_l)*_ s( g, m) + i -( r o _ —i, m)
For a proof, see [82].REMARK. It is not hard to compute that
(42) S(2,m) = ±[%~\ I [%-]—.
For q = 2 and m even (37) yields (29), that is, Petrovskii's first theoremfollows from the first Petrovskii-Oleinik theorem.
For q - 2 and m odd (41) yields
(43) ,)!=
But this estimate is worse than (26) in Petrovskii's second theorem.1 Weconsider the example of a quintic curve C5 consisting of an odd branch Ithat intersects the line x 0 = 0 of RP2 in five consecutive points and sixovals situated in one of the domains formed by x 0
= 0 a n d I, as indicatedin Fig. 10. We may assume that in R2 (x0 = 1) we have C5 > 0 in theshaded domain. It follows from (26) that such a disposition of ovals cannot
Fig. 10
exist (p = 6, / = 0, Δ+ = 5, Δ " = 1, 3 m 2
g~4 m + 1 = 7). Now (43) does not
prohibit such a disposition of ovals, because χ(5 + ) = 6 and 3 m ' + 5 = 10. Ifο
we return to the proof of Petrovskii's theorems in [56], we can see that forThis fact remained unnoticed in [82].
38 D\ A. Gudkov
even m the closure of M+ in RP 2 yields a set B+ for whichΙ χ(Β+) | = | Ρ - L |. For odd w, | χ(Β+) | is less than | ρ -1 + A+ |, that is,(43) is weaker than (27), which is weaker than (28), and consequentlyweaker than Petrovskii's second theorem (26).
We now turn to the space RPq of arbitrary dimension q and constructhypersurfaces f(x0, . . . , xq) - 0 for odd order m just as' in the proof ofPetrovskii's second theorem in [56]. We assume that the polynomial/ ( I , x!,..., xq) =• F{xx, . . . ,xq) satisfies all the conditions indicated above(see b) and the algebraic lemma). As before, we denote the set F > 0 ( F > c) inR 9 (*o = 1) by M+ (Mc). However, in addition we choose a closed ballD with centre at the origin of Rq and a" sufficiently large radius so that thesurfaces F — c = 0 for c 2 < c < cx intersect the boundary sphere of Dtransversally. The numbers cx and c2 are defined in the Petrovskii-Oleiniklemma. It is obvious that such a ball exists. We set M+ Π D = B+ andMc Π D = Bc. It is not hard to see that for B+, Bc and Bc , when theconditions of the Petrovskii-Oleinik lemma hold, then for m odd
(34')
The proof is the same as for (34). Of course, (34') is also valid for evenm. However, for even m it is weaker than (34).
Applying (34') with q = 2 and odd m we obtain Petrovskii's secondtheorem, because χ(5+) = ρ - / + Δ+, x{Bc) = k, and x(BC}) = 1.
5. In 1951 a paper of Oleinik [90] dealt with the generalization of theresults of [56] to algebraic space curves. Let fiixQ, x1} x2, X3) andf2(xo, *i> *2> ^3) be non-singular homogeneous polynomials with realcoefficients, of degree mx and m2, respectively. Suppose that the surfaces/ι = 0, / 2 = 0 , and x 0 = 0 intersect pairwise in non-singular curves andthat all three intersect in k real points without tangency. We introduce thenotation
(44) /*(1, xu xv xa) = Ft(xlt xt, x3) (i = 1, 2).
We denote the set of points in R P 3 for which /\ = 0 by B, the set ofpoints in R 3 (xQ = 1) for which F t = 0 and F2 > 0 (F 2 > c) by M+ (Mc),and finally, the closure of M+ (Mc) in R P 3 by B+ (Bc). The ideas in [82]are applied to this case in [90]. By comparison with [82] an additionaldifficulty arises in [90] in connection with the fact that the critical pointsof F2 in the set Fx = Q satisfy the system of equations
(45) F, = 0, F2X3F1Xi-F2xiFixs = 0,
which is satisfied by the coordinates of the points that are solutions of thesystem of equations
(46) Ft = 0, FiX3 = 0,
The topology of real projective algebraic varieties 39
Under certain conditions these points are not critical for F2 in F, - 0.The main result of [90] is the following lemma:OLEINIK'S LEMMA. Let / ;(x 0, X\, x2, X3) be non-singular homogeneous
polynomials with real coefficients of degree mi (i = 1, 2). Suppose that allthe solutions of (45) are finite (that is, lie in C 3 ) and distinct and that Cjand c2 are constants such that c2 < F2(Pa) < cl for each real criticalpoint Pa of F2 on F^ = 0. Then
,/n\
(47),r> \
χ(Β+)2) 2 —
-γ- (mf — 1) {mi — 2) — C^i+rm2 .!^
Two theorems follow from this lemma.OLEINIK'S FIRST THEOREM. // m2 is even, then
(48) |x(Z?+)K-i/w?-f-|-m1m* + -^m>2 — m\-
PROOF. For even m2 it is not hard to show that x(Bc ) + x(Bc ) =χ(Β).Therefore, (48) follows from (47).
For odd m2 we take a closed ball D in R 3 with a radius so large thatthe curves of intersection of Fj = 0 and F2 - c = 0 for c2 < c < c1
(where c1 and c2 are defined in Oleinik's lemma) intersect the boundaryof D transversally. We set B+ = M+ Γ) D and Bc = Mc C\ D (for thedefinition of M+ and Mc see above).
OLEINIK'S SECOND THEOREM. // m2 is odd, then
, 1 3 _ , |χ(θ) + * |
PROOF. It is not hard to show that (47) remains valid if B+, Bc, and5 C j are replaced by B+, Bc, and 2? In addition, x(BcJ = χ(Β) andx(Bc) = k.Therefore, (49) holds.
REMARK. Petrovskii's first theorem follows from (48) for mx = 1 andeven m2. Petrovskii's second theorem follows from (49) for mx = 1 andodd m2, because χ(Β+) - ρ — I + Δ+ and χ(Β) = 1. Now (49) is strongerthan (18) in [90]. For (18) in [90] yields I %(B+)\ < 3 "* i~^2 + l + ^ i .
for m1 = 1. The right-hand side of this inequality is smaller by 1 than theright-hand side of (27), however, | χ (B+) | < | ρ - I + A+ | - 1; see theexample with the quintic curve. In this survey the proof of Oleinik's secondtheorem corresponds fully with that of Petrovskii's second theorem in [56].
In [90] Oleinik proved that (48) is sharp for curves of arbitrary evenorder on a surface of order mx = 2, by constructing an appropriate exampleby a slight modification of the method in Hubert's paper [10].
40 D. A. Gudkov
§7. Sextic curves in RP2
1. All the logically possible .M-curves C6 were listed in Table 2. In exactlythe same way one can list all the logically possible curves C6 consisting of10 ovals, etc. As a result we obtain the following 68 logically possibletypes of non-singular sextic curves:
Table 3
i . X) i-9 i.7 A4 3.= J_fi ±71 ι 1 Ί 1 '» 1 ύ 7 1^7 1 D ; 1 " ι ι Ι
± 1 A? 3_v LL L* 7
-5- A i JL? ^.^ J_A Β
1 ι 1 ]7 ι ώ7 ι J , 1 t , 0
A i., A2 J-3
1 7 ι 1) ι ' . ^
1 7 ι 1, 3 ^ 1 , 1, ])
JL ,
1 ι L
I0
We denote by (1,1,1) the type of a curve C6 consisting of one nest ofweight 3. We have already seen that curves C6 of type j 9 and of type2 1 can be constructed by Harnack's and by Hubert's method. In the statedment of his 16th problem, Hubert [10] (see [119]) made the conjecturethat of the Λί-curves C6 only curves of types % 1 and j 9 exist.
2. In 1909 Kahn [20] and Lobenstein [21] published papers writtenunder Hilbert's direction. In these papers an attempt is made to provethat no curve C6 of type 11 exists. In Hubert had already stated his convictionthat to solve his 16th problem one must use a continuous change in thecoefficients of the curve (surface, etc.). A method of proving that there isno curve C6 of type 11 is presented in [20] and [21] in accordance withthis general idea. This method is as follows: we assume that Fo = 0 is theequation of a curve C6 of type 11 and that Fo < 0 inside the ovals.
a) We take nine points yx, . . . , y9, one on each of the nine ovals of Fo.Through these points we pass a cubic curve φ. Such a curve exists, isunique, and does not intersect the remaining two ovals α and β
The topology of real projective algebraic varieties 41
of F o . We make a continuous change in the coefficients of Η according tothe formula
Η == (1 _ t)F0 + t^ = 0,
where t is a parameter that increases monotonically on [0, 1]. It is nothard to show that under certain conditions there exists a /*, 0 < /* < 1,such that all curves Η for 0 < t < t* consist of 11 ovals outside eachother and that the curve H(t*) = FY consists of 10 ovals and one isolatedpoint ζ ι from the shrinking of a or β. Similarly {yx Ξ Ζ{), by a continuouschange in the coefficients of F t we arrive at a curve Fi0 consisting of oneoval a and 10 isolated points z1} . . . , z 8 , z 9, z 1 0 lying outside a.
b) We consider the set of sextic curves Η having at least simple doublepoints at z l 5 z 2 , . . . , z 8 . This set is the three-dimensional projective spaceRF 3 . The indecomposable curves having only 10 simple double points(that is, simple rational curves) form a set X in R F 3 , which is the set ofnon-singular points of some algebraic curve in R F 3 . Now X consists offinitely many components. Each component Κ is either diffeomorphic to acircle, (a closed component) or diffeomorphic to an interval (an intervalcomponent). The number of end-points F* of all the interval componentsof X is finite. The point F 1 0 obtained above lies in some component Kl
(of X). We move F10 along Kx so that the oval a shrinks. This is possible.Here zl, z 2, . . . , z 8 remain fixed, but z 9 and z 1 0 move (non-fixed singularpoints). Now Κγ cannot be closed, because a shrinks. Therefore, there mustoccur a complication F1Q^- F* to F*, a decomposable or complexunicursal. For example, F* can have a point of osculation arising from thefusion of a non-fixed singular point of Fl0 with a fixed or two non-fixedpoints.
c) We now assume that we have succeeded in proving that there exists acontinuous broken line without self-intersections F * F 1 0 in R F 3 (this lineconsists of interval components of X, their end-points, and straight-linesegments) such that, as F moves along the line F*Fl0 continuously in thedirection from F* to Fl0, the oval a of F shrinks constantly and
F1Q Ε R F 3 corresponds to a curve F10 consisting of the shrunk oval a.and 10 isolated points z l 5 . . . , z 8 , z*9, z1 0 outside a. Then F 1 0 G K2 is acomponent of X. The process is then repeated. Since a shrinks the wholetime, a moving point in R F 3 cannot return to the components Kl,K2, . . .that have already been traversed. But the number of such components isfinite. Therefore, we obtain a contradiction, which proves that there doesnot exist a sextic curve of type 11.
Kahn and Lobenstein investigated the possible complications F 1 0 =*• F*and achieved some results in the development of the method. However, ontheir own admission [20], [21], they did not prove that there is no curveC6 of type 11.
3. In 1911-1913, Rohn in two articles [23], [25] attempted to prove by
42 D. A. Gudkov
the same method that there is no curve C6 of type γ or 11. He made abig contribution to the development of Hilbert's idea. Therefore, we callthis method of proving that certain types of non-singular curves C6 do notexist the Hilbert-Rohn method. In his papers Rohn did not prove completelythat there are no curves C6 of the types y and 11. The fact is that he didnot deal with all the logically possible cases: which decomposable curvesand complicated unicursals should be included in R P 3 mentioned above?Which of these can be eliminated from RP 3 by a variation of thesingular points zu z2, . . . , zs (see §2)?
4. I mention that in 1907 Wright [18] expressed the same idea as Hubert.In this paper a completely non-rigorous, descriptive attempt is made toprove that there is no curve C6 of type 11. In 1927 Donald [37] repeatedthe same non-rigorous attempt to prove the same theorem. Apparently, hedid not know of the papers of Kahn, Lobenstein, and Rohn. In 1936,Hilton [54] devoted a paper to a criticism of [37].
5. In 1948 Andronov suggested to me that I should study the topologyof real algebraic surfaces and attempt to apply the concepts of roughnessand degrees of non-roughness, as formulated in the qualitative theory ofdifferential equations. On the advice of Petrovskii in 1950, I also studiedthe concrete problem of the disposition of the ovals of non-singular curvesC6 (see [79]). The analysis of the papers of Rohn and then those of Kahnand Lobenstein from the point of view of the concepts of roughness anddegrees of non-roughness turned out to be fruitful. Without a classificationaccording to degrees of non-roughness it was practically impossible toinvestigate the numerous logically possible complications F l o => F* thatarise from applying the Hilbert-Rohn method. As a result I was able toprove some general theorems on the topology of algebraic curves; see [101],[111]-[113], [116], and the account in §1. Next I could develop
the Hilbert-Rohn method to the extent that it became suitable to provethat the types of non-singular curves C6 above the broken line in Table 3do not exist (see [120]).
I recall that the non-existence of a curve C6 of type 11 follows fromPetrovskii's theorem. Curves C6 of types γ 1 and | 9 can be constructedby Hilbert's method (see §4), curves of types | 2 and γ 6 by a slightmodification of Hilbert's method (see Fig. 4), and a curve of type γ5 by amodification of Harnack's method. Hence, all the curves of the types lyingbelow the broken line in Table 3 can easily be constructed, except forcurves of the types γ 5 , γ4, and γ 3 .
I was able to construct curves C6 of types 15, γ4, and γ3 by thefollowing method. It is not hard to show [123] that there exists aquartic curve C4 consisting of four ovals α, β, 7, and δ arrangedrelative to the four lines x0, xx, x2, and L as indicated in Fig. 11.1 (Lpasses through the point of intersection y0 of xx and x2). We choosex0, X], and x2 as fundamental lines of a projective coordinate system in
The topology of real projective algebraic varieties 43
the x-plane of Fig. 11.1. We denote by >>;- the point of intersection of x,·and xk (i, j , and k are pairwise distinct, i, j , k = 0, 1, 2). We perform the
Fig. 11 u0 = - 1 6 .
quadratic transformation x 0 = yiy2, -χι = y^y-i, χ2 = }Ό>Ί °f t n e x-into the y-plane (see Fig. 11.2). By the properties of quadratic transforma-tions (see [136*]), L is carried into a line L passing through x 0
(the point of intersection of yx and y2)· Now C4 is carried into a curveC5 consisting of an odd branch a with a node at x 0, a figure eight β withnodes at Xj and x2, and two ovals γ" and δ, arranged relative to L and αas indicated in Fig. 11.2. It is not hard to see that we can shift L into aline Ex and simplify x 0 , xx, and x 2 of C5 so that we obtain a simplecurve E^Cs; see Fig. 11.3. It follows that curves C6 of types j 5 , | 4 , and13 exist. Thus, the disposition of the ovals of a non-singular curve C 6 isknown completely: the non-singular curves C6 that exist are precisely ofthose types that lie below the broken line in Table 3.1 I mention that theexistence of a curve C6 of type 15 was unexpected. Hubert had assumedthat such a curve does not exist; see the formulation of his 16th problem1 In [101] I asserted incorrectly that curves C6 of types -j5, j4, and γ3 do not exist. I could not publish
the corresponding wrong proof. These curves are constructed in [121], [123], [122].
44 D. A. Gudkov
in [15].The method used to investigate the disposition of the ovals of curves C6
is not applicable to curves of higher orders. However, the investigation ofsextic curves has led to interesting results (in addition to the very solutionof the problem on the disposition of the ovals of curves C6).
Firstly, from the decomposable curve EiCs of Fig. 11.3 one can con-struct a series of Af-curves in exactly the same way as from the curve ofFig. 5.4 in Harnack's method. A new series of .M-curves is obtained. Thisseries contains, in particular, a curve Cm for each even m > 6 consistingof ovals a and β and 3 w ' ~ 6 m - 5 more ovals lying outside each other (along
8
with a and (3); five ovals lying outside each other and inside a; and8
ovals lying outside each other and inside β. That is, Cm has the type
(50)_ -5-m(m — 6)5 8
Secondly, one can easily show that the curve E\CS in Fig. 11.3 cannotbe constructed by the small parameter method from curves of lower orders.It is entirely possible that for each m > 6 there exists a decomposablecurve Ek Cm_k that cannot be constructed by the small parameter methodfrom decomposable curves of lower orders and is the beginning of a newseries of Af-curves. This points to the extraordinary difficulty of construct-ing all the existing A/-curves by the small parameter method.
Thirdly, from the results of investigating the curves C6 one can formulateone of the necessary conditions on the disposition of the ovals of .M-curves.Let Cm be a non-singular curve of the mih order in R P 2 . We denote thenumber of even ovals of Cm by Ρ and the number of odd ovals by L. Itis apparent from Table 3 that the congruence Ρ — L = 1 mod 8 holds forAf-curves C 6. Therefore, it is interesting to investigate Ρ - L mod 8 forM-curves constructed by the standard methods ( § § 3 and 4). For example,Ρ - L = (m/2)2 for .M-curves Cm (of even order m) constructed by Wiman'smethod (§4, Wiman's theorem). For the curves (50) we haveΡ — L = (m/2)2— 8. Both of these equalities can be combined into thecongruence
(51) Ρ - L == {m/2)2 mod 8.
All of the .M-curves of orders 8, 10, and 12 constructed by the knownmethods satisfy (51). It is not hard to see (see the beginning of §6) thatthe .M-curves constructed by Harnack's method also satisfy (51). The listof such examples can be continued. Therefore, I made the following con-jecture: (51) holds for M-curves of even order m; see [123]. Soon, Arnol'dproved that (51) holds mod 4 for such curves; see [124]. Then Rokhlinproved the conjecture in full; see [125], [126].
Fourthly, the investigation of curves C6by the Hilbert-Rohn method madeit possible to give a meaning to and prove certain theorems on the
The topology of real projective algebraic varieties 45
simplifications and variations of the singular points of curves of the mthorder (§1).
The problem of investigating non-singular quartic surfaces is quiteinteresting (see § 11). Hubert also pointed out this problem in stating'his 16th problem.
§8. Arnol'd's theorems
In 1971, Arnol'd [124] showed how to apply, modern topological methodsto the investigation of the topology of real algebraic varieties. Here is hismain result:
ARNOL'D'S THEOREM ON Λί-CURVES. Let A be an M-curve of evenorder m - 2k in RP2. Then
(52) Ρ - L = k2 mod 4,
where Ρ is the number of even and L the number of odd ovals of A.I sketch a proof of this theorem.1) Let /(x0, Χι, x2) be a real non-singular homogeneous polynomial of
degree 2k. Here [x0: xx: x2] are point coordinates in CP2. Suppose thatthe equation / = 0 determines our M-curve in RP2 and a complex curveCA in CP2.
2) DEFINITION OF THE MANIFOLD Y. The manifold Υ is a doublecovering of CP2 ramified along CA. We denote its projection byρ: Υ -»• CP2. Then Υ is a four-dimensional, smooth, orientable manifold(without an edge). We assume that CP 2 and Υ have the natural orientation.
3) DEFINITION OF THE INVOLUTION τ AND THE FORM Φ τ . Wedenote by r: Υ -*• Υ a smooth non-identity automorphism of ρ: Υ -* CP2.
IT /-y i?\
Next, we set H2(Y) — • 2rpor' (in fact, there is no torsion). The inter-section index (a, b) of the classes a, b Ε H2(Y) determines on H2(Y) asymmetric, unimodular bilinear form (Poincare duality). Letτ*: H2(Y) -*· H2(Y) be the isomorphism induced by r. We set(τ, a, b) - Φ > , b).
4) DEFINITION OF THE MANIFOLD Π. Let B+ be the set of points(in RP2) where / > 0, and suppose that the sign of / is chosen so thatB+ is orientable. Then
(53) χ(Β+) =P-L.
We denote by conj: CP 2 -*• CP2 the involution of complex conjugation.This involution is covered by two smooth involutions of Υ that aretransposed under τ. Of these two involutions we denote by conj: Υ -*• Υthe one whose set of fixed points is p - 1 ( 5 + ) = Π. Now Π is two-dimensional, smooth, orientable, and smoothly embedded in Y, and rchanges the orientation of Π.
46 D. A. Gudkov
5) DEFINITION OF THE SURFACE D + B+. Since A is an M-curve,it divides CA into two halves (see Klein's and Hurwitz's proof of Harnack'stheorem, §3). Let one of these halves be D. The surface D + B+ defines atwo-dimensional cycle in CP2 with coefficients in Z2. Now D + B+ is com-pact, without an edge, and, possibly, non-orientable.
6) It is proved in [124] that the class w = [G4] € H2(Y) is canonical(see [147*]) for Φ τ , that is,
Φτ(χ, χ) = Φχ(χ, w)mod2, Vx£H2(Y).
If x' G H2{Y), it is not hard to see that w' = w + 2x' is also canonicaland that
(54) Φ τ (wr, w') = Φ τ (w, w) mod 8.
7) It can be proved that
(55) ( [G4], [CA]) = 2k2 and ( [Π], [Π]) = - χ(Π),
where ΙΠ] is the class of Π in H2(Y) Hence from (54) we have
(56) am,because Π is a duplicate of B+ (the two copies of B+ merge along A).
8) ARNQL'D'S LEMMA. The homology classes [C4] and [Π] coincidemod 2.
The proof is based on the fact that the number of points of intersectionof D + B+ with the line x0 = 0 has the same parity as k (see [124]).
9) By Arnol'd's lemma and 6), w' = [Π] is canonical for ΦΤ. Therefore,(55) holds for w'. We note that T^U? = w, where w = [CA], because CAconsists of the fixed points of τ and T*W' = — w\ hence Φ τ (w,w) - (w,w)and Φτ (w ',w ') = -(«; \w'). From (54) we have{[CA), [CA]) + ([Π], [Π]) Ξ 0 mod 8. The assertion (52) of the theoremnow follows from (55), (56), and (53).
In [124] there is also a new proof of Petrovskii's first theorem. We notethat τ* and conj* commute on H2{Y) and form a Klein group of order 4:{1*, τ*, con]*, -ts-conj*} = G. We introduce the form <$>h{a,b) = {h* a,b),where a, b G H2{Y) and h* e G.
Arnol'd first deduces the following table: T , , ,
Tr A* = trace of Λ*
sign Φ^ = signature of Φ^
1*
2 —2fe2
con]*
2(P — L) — 2
2(L-P)
t *
-2g
2fc2
τ* conj*
2{L-P)
-2+2(P-L)
The topology of real projective algebraic varieties 47
Here g is the genus of G4. See also [148*].Next, Arnol'd proves Petrovskii's first theorem. We consider the linear
space Μ = H2(Y, R) with the scalar product defined by the intersectionindex. This product is non-degenerate and is preserved by conj*. Therefore,Μ decomposes into an orthogonal direct sum of eigenspaces Mx and ΜΛ
corresponding to the eigenvalues of conj*. Each of Mx and ΜΛ is the directsum of orthogonal subspaces on which the scalar square is positive (negative)definite: Mx = M\ + M\ and M_x - Μ*Λ + MZ\ • We denote the dimensionsof Mj , Ml, Μ*ι, and M'_x by a, b, c, and d, respectively. It is then obviousthat
Trconj* =^a-rb — c — d, Tr l* = a + 6 + c + d,
sign <Pconj = a — b — c + d, sign Φχ = a — b + c — d.
Taking Table 4 into account, we have b + d = 3k2 — 3/c + 1 andb - d = 2(P - L) - 1. Since b and d are non-negative, \ b - d \<b + d,that is, I 2(P - L) - 1 | < 3k2 - 3k + 1. This is Petrovskii's first theorem(see §6).
Arnol'd denotes the number of even ovals of A that externally boundthe component of B+ with positive, zero, and negative Euler characteristicsby P+, Po, and P_, respectively. Similarly, L+, Lo, and Z,_are the numbersof odd ovals that bound (externally) the component of B_ with positive,zero, and negative characteristics, respectively.
ARNOL'D'S THEOREM ON NESTS. For a non-singular curve A of orderm = 2k in RP2
(57) P ( k \ l k 2 \ ^ k l ^ k 2 \
(58)
EXPLANATION. Let 7,· be an even oval of A. We denote by Π,· thecomponent of Π lying in Υ over the component of B+ bounded fromwithout by 7,·. It is obvious that conj* [Π] = [Π]. Arnol'd remarks thatthe first of the inequalities (57) follows provided that the classes [Π,-]corresponding to the oval 7, bounding from without the components ofB+ with negative and zero characteristics are linearly independent, because
in this case P_ -j- Po ^ dim M* = ^ . Recently, in Arnol'd's seminar,
Rokhlin reported that the requisite linear independence holds.The rest ofArnol'd's inequalities are proved analogously.
In [124] Arnol'd makes the assumption that the following inequalityalways holds for an Af-curve of even order m = 2k:
(59) P + + L + ^ + ±
This is false. By Wiman's method one can construct a curve of order 12having outside each other: one nest of weight 4, eight nests of weight 2,
48 D. A. Gudkov
and 36 empty ovals. For this curve P + + L+ - 45 < 46.
§9. Harnack's theorem for varieties and Rokhlin's theorems
1. Let A be a non-singular algebraic curve of order m in RP 2 defined bythe equation / = 0. Harnack's theorem [7] yields an exact estimate for thenumber of components of A. It is not hard to restate Harnack's theoremin the form of an exact estimate of one of the following numbers: the sumof the Betti numbers of A, the sum of the even-dimensional (odd-dimensional)Betti numbers of A, or the number of components of the complementRP 2 -A. If m is even, the sum of the Betti numbers of B+(f > 0) can beadded. Therefore, there are many logical possibilities of generalizing Harnack'stheorem.
2. It is appropriate here to mention an idea which Arnol'd has communi-cated to me. Let Ζ be a compact Riemannian manifold. We denote byλι < λ2 ^ . . . the eigenvalues of —Δ (where Δ is the Laplace operatoron the set of functions on X). Each eigenvalue is written as many timesas its multiplicity indicates. A function φ on X is said to be subordinate tothe k-th eigenvalue \k if φ is a linear combination of the eigenfunctionsfor \χ, λ 2 , . . . , Xk. The following theorem is in [149*]:
COURANT-HERMANN THEOREM. Suppose that the function φ: Χ -»• Ris subordinate to the eigenvalue \ . Then the set of zeros of divides X intomore than s parts.
Suppose that the homogeneous equation f(x0, * i , . . . , xq) - 0 ofdegree m with real coefficients defines a hypersurface A in RPq. We con-sider the sphere 5« = {(x0, xlt . . ., xq) € R9+1 Κ + · · -+4 = 1}· I f m
is even, then / defines an even function /: Sq -*• R, and hence a functionψ: RPq -*• R. Now / is the sum of a spherical function of degree m and alinear combination of spherical functions of smaller even degrees. Therefore,ψ is subordinate to the eigenvalue Xr+l of the Laplace operator of thestandard metric on RPq, where r is the dimension of the space spanned byall even spherical functions of degree less than m. A calculation shows thatr =(q + m — 2\. Hence by the Courant-Hermann theorem
\ Q I
(60) p°(RPq-A)<^l + (ci + m ~ 2\
where p°(K) is the number of pieces of K.The case of odd m is analyzed similarly, the only difference being that
φ is not a function, but a section of a one-dimensional fibering over RPq.Arnol'd has reported (see [135]) that the Courant-Hermann theorem is
false in general. However, (60) is sharp for q = 1 and q = 2 (for q - 2 itis attained on m lines in general position). In all cases known to me (60)holds. Therefore, (60) may be regarded as Arnol'd's conjecture.
According to a remark by Arnol'd [135], to prove (60) it suffices to
The topology of real projective algebraic varieties 49
prove the Courant-Hermann theorem for a sphere with the standard metric.3. Until recently there were four papers [58], [92], [114], [115]
devoted to finding estimates for the Betti numbers of a real projective alge-braic variety. I do not present here the estimates for algebraic varieties inR^, which can be found in the papers just mentioned.
In 1939 Bieberbach [58] gave the roughest estimate for the number ofcomponents of a real algebraic variety A that is the intersection of hyper-surfaces of orders πι1} . . . , ms in RPq. He proved that the number ofcomponents p°(A) does not exceed some number that depends ontnx, . . . , ms but not on the coefficients of the hypersurfacesdefining A. The proof uses only elementary arguments and proceeds byinduction on q.
In 1949, Oleinik [81], [92] investigated the case of a non-singular hyper-surface A of order m in RPq. Let H*(A; Z 2 ) be the total homology spaceof A with coefficients in Z 2 . Then dim H*(A; Z 2 ) coincides with the sumof the Betti numbers of A with respect to Z 2 . We denote byΣ2(Α) (Σι(ν4)) the sum of the even-dimensional (odd-dimensional) Bettinumbers of A with respect to Z 2 . We tabulate the estimates obtained byOleinik. The number s(q, m) was introduced in §6 (see (32)). For q = 2the estimates in Table 5 are not sharp. The proofs use (just as in [82])Morse theory and the Euler-Jacobi theorem for polynomials. There are someother estimates in [92].
Table 5
dim II* (A; Z.>)
9 even
in even
, ( r a - l ) ? + M m - l )
^ 7rt-2 '
, 9(9 + 3)
' 2
Σ i =-- -77- dim II %
m
< «
μ;
odd
.-,„
z2)
r ?
9 odd
m even
_ ( m _ l ) e + i _ ( w i _ l ,
"*~ m — 1 '
, 9(9 + 3)' 2
<(m —1)5 —s(9, m) +
1 m - 2 !
9(9 + 3)1 2
m odd
- s (9, TO) +
In 1964, Milnor [114], using the apparatus of Morse theory, found anestimate for dim H*{A; Z 2 ) under the assumption that A is the intersectionof hypersurfaces of orders my, . . . , ms and can have arbitrary singularities.This estimate has the form
(61) (A; Za) <g-m(2m - I) 7 ,
50 D. A. Gudkov
where m = max {m^ . . ., ms}. If A is a non-singular hypersurface, thisestimate is asymptotically worse than those in Table 5.
In 1965, Thom [115] examined the following case. Let A be an algebraicvariety of dimension η < q —2 in RPq defined by a single equation / = 0of degree m, where / > 0 in the whole of RPq. Then
(62) dimH,(A; Z 2 ) < ( m — i)q-n + m^"*1 + .,.+ηι"-1.
This estimate is also obtained by the apparatus of Morse theory. In Thorn'spaper there are many necessarily unsharp estimates of dim H*(A; Z 2 ) forA in Κ".
At the end of [115] is the following argument. Let CA be a non-singular hypersurface of order m defined by an equation with realcoefficients. Let A be the real part of CA, and let conj: CA -*• CA be com-plex conjugation. Then A is the set of fixed points of this involution.Thom remarks that according to Smith theory (see [150*])
(63) dimΗ,(Α; Ζ 2 ) < dim H% {CA; Z2).
It is important that the homology of CA is known. Hence Thom obtainsthe following estimates for a non-singular hypersurface A or order m in
(64) dim/M^X _± h g + 1 if q is odd.
Now (64) for q = 2 yields Harnack's theorem. Thom did not notice this.He also remarks that (63) can be applied to the case when A is a regularintersection of hypersurfaces. In this case the homology of CA is alsoknown.
4. The remaining part of this section gives an account of Rokhlin'sresults in [126], [130]. In [125] Rokhlin proves the congruence (51) fora plane Λί-curve of even order m. The proof is rather complicated. In [126],[130] two problems are solved.
1) Harnack's theorem is generalized to varieties of dimension η > 1.There is, firstly, an indication of Thorn's generalization in [115], andsecondly, another generalization of Harnack's theorem is proved.
2) The congruence (51) for ΛΖ-curves is generalized to varieties of arbitrarydimension η > 1. Here a proof of (51) for M-curves in R P 2 , is obtainedsimpler than that in [125].
Rokhlin uses some results from Smith theory. We begin with these results.THE SMITH SEQUENCE. Let X be a smooth manifold, τ a smooth
involution on X, and X' the quotient space Χ/τ. Suppose that the set Φof fixed points of τ is regarded as a subset of X'. Then there exists asequence of homomorphisms
The topology of real protective algebraic varieties 51
(65) . . . --^i ΗΓ (X\ Φ; Z2) 0 ΗΓ (Φ; Z2) X Hr (Χ; Ζ2) Λ
— # r (Χ' Φ; Ζ2) Λ # r _ t (Χ', Φ; Ζ2) φ # r _ t (Φ; Z2) - ^ . . . ,
having the following properties:
1°. 77ie component Hr(X', Φ;Ζ2)<\χΗ^_ι (Χ1, Φ; Z 2 ) 0/7,. is Af/wied ftj
ί/ze formula ξ -> ξ Π x, where χ is the characteristic class of the coveringX - Φ -»· X ' - Φ <fe/med 6>» τ.
77ie component Η/.(Χ',Φ;Ζ2)ΧΗι._ι ( Φ ; Ζ 2 ) of yr is a boundary homo-morphism.
2°. The automorphism Hr(X; Z2) 1? Hr (X; Z 2 ) induced by τ is theidentity on I m o , .
3°. The sequence is exact.For the remaining data on (65) and the proof, see [ 1 5 0 * ] , [ 1 5 1 * ] .
SMITH'S INEQUALITY. For the spaces X and Φ in the Smith sequence
(66) dim Η, (Φ; Za) < dim Η*(Χ; Ζ2).
PROOF. Since (65) is exact, it follows that
(67) dimHm(X; Z2) = dim # * (Φ; Z2) + 2 2 dim (ker yT),
therefore (66) holds.For the rest of this survey we denote by CPq, the complex projective
space of dimension q by RPq the real part of CPq (which is a q-dimensional real projective space); by [x 0: x^. . . . : xq ] the coordinates ofa point in CPq, and by conj: CPq -> CPq the involution of complexconjugation; finally, by χ{Μ) and a{M) the Euler characteristic and thesignature (if it exists) of M.
5. By a non-singular algebraic variety CA in CPq we mean a compact,complex-analytic subvariety in CPq that is invariant under conj: CPq -+CPq.The restriction of conj: CPq ->• CP 9 to C4 is also written as conj: CA^-CA.Now CA is orientable. We assume that the natural orientation has beenintroduced. The real part of CA, which is a variety, is denoted by A. If weset dim A = n, then dim CA = In. For the so defined varieties A and CAwe have the following theorem:
THE HARNACK-THOM THEOREM. The following inequality holds:
(68) dim HM\ Z2) < dim Η,(0Α; Ζ2).
PROOF. We write down the Smith sequence (65) for the caseX = CA, τ = conj, and Φ = A. Then (66) becomes (68).
We note that in this case it follows from (67) thatdim H*(CA; Z 2 ) — dim H*(A; Z2) is an even number, which we denote by2t. From Thorn's remark and (64) we know that for plane curves (68)yields Harnack's theorem. Therefore we make the following definition:
DEFINITION OF AN (M-t)-MARIETY. Suppose that
52 D. A. Gudkov
dim H*(A; Z 2 ) = dim H*(CA; Z 2) - 2t. Then A is called an (M - t)-variety.In particular, for t = 0, A is called an M-variety.
EXAMPLE. For an irreducible non-singular curve A in RP 3 the Harnack-Thom theorem yields dim H* (A; Z 2 ) < 4~x(G4) = 2 + 2g, where g is thegenus of CA. If A is situated on a quadric, then the estimate m theHarnack-Hilbert theorem (see §5) follows.
Our next aim is to prove Rokhlin's theorem on M-varieties. We denote byu(X), where Ζ is a smooth closed manifold of even dimension 2k, thek-dimensional homology class By of X, that is, the elementu £ Hk(X; Z 2 ) for which (a, ξ) = (£, ξ) for any ξ G Hk(X; Z 2 ). The ele-ments υ of the integral group Hk{X) that are carried into u(X) under thecoefficient homomorphism Hk{X) -*• Hk(X; Z 2) are called the integralhomology classes By.
ROKHLIN'S FIRST LEMMA. If η is even, then CA {of dimension 2ri)has an integral homology class By υ for which conj*u = (-1)"^2υ.
A more general proposition, from which this lemma follows, is proved in[130]. In [126] the lemma is proved for a special case.
ROKHLIN'S FIRST THEOREM. Let η be odd. Then from
(69) dim H,(A> z s) = d i m H^CA; Z2) = %(CA)
it follows that
(70) %{A) = o(CL4) mod 16.
PROOF. 1) We consider the group Hn{CA), which is well known to befinitely generated free Abelian. We denote by G+ the subgroup of Hn(CA)consisting of the elements that are invariant under the automorphismconj*: Hn(CA) -»• Hn(CA). The elements ξ Ε Hn(CA) for whichconj* ξ = - % also form a subgroup, denoted by G_.
2) We write down the Smith sequence (65) for the case X = CA,τ = conj, and Φ - A. Then it follows from (69) that in (67) for this caseKer yn = 0 and that, by the properties of (65),conj*: Hn(CA; Z 2 ) -*• Hn{CA; Z 2 ) is the identity automorphism. It followsthat the classes ξ — conj*ξ and % + conj*ξ are divisible by 2 in Hn(CA) forany ξ € Hn(CA), that is, the formula
ξ = j [I + conj» %] + -j[l — conj* ξ]
effects a direct decomposition Hn(CA) = G+ © GL, which is obviouslyorthogonal with respect to the intersection index.
3) Let Q be the quadratic form of the self-intersection index onHn(CA), and let Q+ and Q_ be the quadratic forms induced by Q on G+
and G_. We denote the signatures of Q+ and Q_ by σ+ and σ_. It is clearthat I det Q+ | = | det Q_ | = 1 and that σ+ + a. = a(CA).
By virtue of the Atiyah-Singer-Hirzebruch formula (see [153*], §6), the
The topology of real projeetive algebraic varieties 5 3
normal Euler number of A in CA is equal to the signature of the quadraticform (£, conj* E), that is, σ(£, conj*£), a n d since multiplication by / carriesthe tangent fibering of A into the normal one, this number is equal to(-1)"/2XG4). Therefore,
(71) ( _ 1 ) η / 2
χ μ ) = σ(ξ, conj,|).
It is not hard to see that σ(£, conj*!;) = σ+ -σ_, hence,
f a(CA) -2a. if η Ξ 0mod4,(/2) χ.(4) = | σ ^ + 2 f f + i f „ = 2 mod 4.
4) We now take into account that for η = 0 mod 4, by Rokhlin's firstlemma, there exists an element .Sy, ν ζ G+, hence Q_ is even. Consequently,a_ is divisible by 8 by a standard lemma; see [147*]. If η = 2 mod 4,υ e G_ and Q+ is even, that is, o+ is divisible by 8. In both cases (70)follows from (72), and the theorem is proved.
6. We now turn to the proof of one more variant of the generalizationsof Harnack's theorem, which was suggested by Rokhlin [126].
Let f(x0, . . • , xq) be a real homogeneous polynomial of degree m, andsuppose that the hypersurface / = 0 intersects regularly in CA a non-singular algebraic variety CB in CPq. We call CA a regular intersection ofCB and f. We denote by A and Β the real parts of CA and CB, and wekeep the notation dim A = n.
For even m let Υ be a ramified double covering of CB with ramificationalong CA. We denote its projection by ρ: Υ ->• CB. Now CB, CA, and Υare orientable. We assume that the natural orientation has been introducedin them.
If A is non-empty and m is even, the subsets of Β defined by / > 0and / < 0 are denoted by B+ and B_ Then 5 + and B_ are subvarieties ofΒ with the common edge A. The involution conj: CJ5 ->• C5 is covered bytwo smooth involutions of Υ that are transposed by the non-identitycovering automorphism p. For one of these involutions the variety of fixedpoints lies over B+, for the other over B_. We denote these involutionsby conj+ and conj". Finally, we set ρ~λ(Β_) = F_ and p'1(B+) = F.
+ •
ROKHLIN'S FORMULA. Let A be non-empty and m be even. Then
(73) dim H* (F_; Z2) = dim H, (A; Z2) -{- 2a_,
where a_ is the dimension of the intersection of the kernels:
the endomorphism H*(B_, A; Z2)-X-Hif(B_, A; Z2) defined byξ ι—* ζ Π ω, where
{ 0 if m = 0 mod 4,
wi (B\A) if m = 2 mod 4,
boundary homomorphism H*(B_,A; Z2)\ H*(A; Z2).
(74) { d i m
[ dim
54 D. A. Gudkov
To prove this we write down the Smith sequence (65) for X = F_ andthe involution r, the restriction of ρ to F_. Then X' = F_/T = B_and Φ = A.
Now (67) yields (73), where a_ = 2 d i m (kei-γ,.), and the yr are the homo-r
morphisms in (65) for this case. Hence, by property 1° of the Smithsequence, a_ is the number indicated above, where ω coincides with the χin 1°. The same formula as (73) holds for dim H*(F+; Z 2 ), but with a_replaced by a+ which is defined similarly in terms of B+ and A.
THE HARNACK-ROKHLIN THEOREM. Let A be non-empty and m beeven. Then
dim Η„ (A; Z2) + 2o_^dimi/^ (Y; Z2),
(Y; z2).PROOF. We write down (65) for X = Υ, τ = conj", and Φ = A. Then
(67) yields the first of the inequalities in (74). The second one is provedsimilarly.
We mention that from (66) for the case under discussion and from (73)it follows that dim H*(Y; Z 2) - dim H*(A; Z 2) is even.
DEFINITION OF AN (Af-i)-PAIR. Let A be non-empty and m be even.We call a pair of varieties (B, A) an M-pair if equality holds in at least omof the relations (74). We call (B, A) an (M - \)-pair if it is not an M-pairand if in at least one of the inequalities (74) the left-hand side is smallerby exactly 2 than the right-hand side. The concept of an (M - t)-pair{B, A) is defined similarly.
EXAMPLE. For η = 1 or m Ξ 0 mod 4 we have Π ω = 0 anda_ = dim (ker 9) = dim H*(B_; Z 2 ) - \ dim H*(A; Z 2 ). In this case theHarnack-Rokhlin theorem yields
2 d im#„(£_; Z 2 )<dim Η, (Υ; Z2),1 2 d im#„(#+; Z 2 )<dim Η, (Υ; Z2).
ROKHLIN'S SECOND LEMMA. For odd η the variety Υ has an integralhomology class By υ for which conj;(υ) = (- l)("+1>/2 v.
The lemma follows from a more general proposition proved in [130]. Aspecial case is analyzed in [126]. Of course, the lemma is also valid forconj*.
ROKHLIN'S SECOND THEOREM. Let η be odd, m even, and A non-empty. Then from
(76) 2a_ + dim Hm(A\ Z2) = dim Hm{Y\ Z8)
it follows that(77) 2X(5_) ES σ(Υ) mod 16.
The proof is quite similar to that of Rokhlin's first theorem, with G4and conj replaced by Υ and conj".
7. For complete regular intersections the generalized Harnack theorem
The topology of real projective algebraic varieties 55
and the Rokhlin theorem can be made more specific. Suppose that we aregiven real homogeneous polynomials fi(x0, . . . , xq), . . . , fs(x0, . . . , xq)of degrees m1, . . . , ms in q + 1 variables. We denote by CA (A) the subsetof CPq (RPq) defined by the system of equations
(78) J^XQ, . . . , Xq) = 0, . . . , fs-i(x0, • • . , Xq) = 0, fs(x0, . . ., Xq) =0.
Suppose that (78) has no singularities on C4. Then CA is called a completeregular intersection of the hypersurfaces (78) (Case A).
In addition, we shall examine the complete regular intersection CB of thetruncated system of hypersurfaces
(79) fi(x0, • • ·, Xq) = 0, . . ., / s _ i ( z 0 , • • ., Xq) =0.
In this case we assume that fs = 0 is of even order ms and intersects CBby CA regularly (Case B).
Of course, a complete regular intersection is a special case of a non-singular algebraic variety (see §9.5). We assume that in Case A all the con-cepts and notation of §9.5 have been introduced and in Case Β theconcepts and notation of §9.6. For complete regular intersections thenumbers on the right-hand sides of (68)-(70), (74), and (77) are concretepolynomials in q, m 1 , . . . , m s (see [126]).
1) Computation of χ(Ο4). We define the polynomials xf(m 1 ; . . . , ms)by the following formulae:
(80)
r q+l if s = 0,i, . . . , ms)=i
[ mu . . . , ms if 0 < s < q,
ms-u ms) = nisXlli (mu . . . , ms_i) —
- ( m . - l J x r C n i . · · · , ms) if 0 < s < q.
Regarding CA (ms is arbitrary) as an ms-sheeted ramified cyclic covering ofa hyperplane section (x = 0) of CB with ramification along a hyperplanesection of CA itself, we can see by induction on q that
(81) x(C4) = x2( m i , . . . . ro.).
2) Computation of dim //*(C4; Z 2 ): It is known (see [152*]) that theintegral homology group Hr(CA) is trivial for odd r Φ η and isomorphic toΖ for even r Φ n. Therefore, we have
Xjii»!, ..., ms) if η is even,
+ l ) — il(mu . .., ms) if η is odd.
3) Computation of a(CA) for even n. We define the polynomials(ή.(ηιι, . . . , ms) by the formulae
(82) dimHm(CA;Zi)=i
56 D. A. Gudkov
1 , i f * - Ο ,σ? (TO^ ..., ms) = \
I mu . . . , ms, if 0(83)
< s = q,
Σ σί7ί(™ι> ·•·> ms_1? μ — 1 , μ) if 0 < s < q.1μ=1
The signature a(CA) can be computed in succession by applyingHirzebruch's "functional equation" (see Theorem 9.3.1 of [154*]). Thecomputation yields
(84) σ (CA) = σ* (mit ..., m3).
4) Computation of dim*(Y; Z 2 ) for odd n). We write down the Smithsequence (65) for X - Υ and for the involution r, the smooth non-identityautomorphism of the covering ρ: Υ -•> CB. Then X' = CB and Φ = CA.Using the fact (see [152*]) that the inclusion homomorphismHr(CA; Z 2) ->• Hr(CB; Z 2 ) is an isomorphism for r < η and is trivial forr > n, we obtain (for yr in the Smith sequence for the case underdiscussion) that
0 for r Φ η + 1,
Hence, by (67) we have
dimH,(Y; Z2) = dim//"» (CA; Z2) + 2 d i m # n + 1 (CB; Z2).
Since /? is odd, we obtain from (82)
(85) dim H. (Y; Z2) = 2χ?_1 ( m i , . . . , me_0 -
— χ? (mi, · . . , m s )^6«(m,, . . . , m s).
5) Computation of a(F) for odd « and even ms. We denote by Ε thevariety in CPq+1 defined by the system of equations (see (78))
/i(x 0, . . . , xq) = 0, . . . , / s_!(x0, · · • , xq) = 0, / s(x 0, •••> ^ ) + V * i = °-We can regard Ε as an ws-sheeted ramified covering of CB with ramificationalong CA and, consequently, as an ms/2-sheeted ramified cyclic coveringof Y. Applying the signature formula due to Hirzebruch [155*] to Ε -*• Υ,we obtain from the Atiyah-Singer formula [153*], §6
(86) σ(Υ)--= 2 ch(ms/2)or2k+i(m,i, . . . , m ^ x ' h , . . . , m . ) y
ft=0
where cfc(m) is the coefficient of t2* in the power series expansion of
t h e CT'(mi' · · · > ms) a r e computed by (83).These computations make it possible to examine the following concrete
examples:1°. For complete regular intersections the Harnack-Thom theorem asserts
that
The topology of real protective algebraic varieties 57
ti(mu . . . , m,), if η is even.
2 (rc-fl) — χ? (mj, . . . , ms) if η is odd.
a) For plane curves (87) yields Harnack's theorem.b) For a non-singular surface A of order ml in R P 3 (87) yields the
estimate
(88) dim Η*(Α; Z2) ^ m\ — im\ + 6m!.
I note that there exist surfaces of orders mx < 4 for which equality isattained in (88). For ml = 1, 2 this is obvious, and for m1 = 3, 4 see §11.
c) For a hypersurface of order m in RP^ (87) yields (64).2°. For complete regular intersections Rokhlin's first theorem asserts that
for even η it follows from
(89) d i m / f ^ ; Z2) = yp(jnx, . . ., ms)
that(90) χ (Λ) Ξ=σ« (/»!,..., ms) mod 16.
a) If q - 3 and s - 1 (a surface v4 of order mj in RP 3 ) this meansthat from
(91) dim.Ht(A; Z2) = m3
1 — bm2
1 + Qml
it follows that
(92) χ (A) = j (Ami — m\) mod 16.
When mx is even, (92) becomes
(93) χ(4) = 0 mod 16.
For Λί-surfaces of the fourth order I had stated (93) as a conjecture(see [127]).
3°. For complete regular intersections (ms even) and odd η the Harnack-Rokhlin theorem asserts that
(94) dim Η\ (A; Z2) + 2a.^b1(ml, ..., ms).
a) If q = 2, s = 1 (plane curves), mi is even, and we assume that B+ isorientable, then B_ is non-orientable. We have
dimHm(B., Z2) = i d i m # , (4; Z2) + 1 = i
where Ρ is the number of even and L the number of odd ovals of A. From(85) and (80) we obtain that dim H*(Y; Z 2 ) = m\ - 3mx + 6. Therefore,(RP 2 , A) is an Λί-pair if and only if A is an Λί-curve.
4°. For complete regular intersections (ms even and A non-empty)Rokhlin's second theorem asserts that for odd η from
58 D. A. Gudkov
(95) dim Η. (A; Z2) + 2a_ = 6« (mlf . . ., ma)
<t follows that
(96) 2χ (Β.) = τ« (m1? .. ., m.) mod 16.
a) In the case of plane curves of even order m t we have σ( Υ) = 2 - ^γ.
Therefore, if ( R ? 2 , A) is an Λί-pair, that is, if A is an M-curve, thenΡ - L = (m1/2)2 mod 8, and (51) is a very special case of Rokhlin'ssecond theorem.
b) For q = 3, s = 2, and m2 even (curves on a surface of order ml inRP 3 ) Rokhlin's second theorem asserts that if
dim #*(£_; Z 2 ) = r^(2m2
1 +m1m2 + m\ -Zm^ -4m2 + 12), then
X(BJ = "± (16 - 4ml - 3/n!) mod 8.
REMARK. The problem arises of searching for an analogue to Rokhlin'ssecond theorem when η = dim A is odd and ms is odd. We consider theexample of an M-curve C7 (of order seven) in RP 2 . Such a curve consistsof an odd branch 1 and 15 ovals and cannot have a nest of weightgreater than 2 nor more than one nest (from the conditions of intersectionwith a line). Therefore, the following 15 M-curves of order seven arelogically possible (the branch 1 is "taken out" for the table):
14 13, 12 11 10. 9 . 8 1 6 8 5 q 4 3 2 i_ - -1 Τ' Τ ' Τ ' Τ ' Τ ' Τ ' Τ Τ ' Τ ' Τ ' Τ ' Τ ' Τ ' Τ '
Of these curves, Ιγ 13 and 1 1 5 can be constructed by Harnack's method,ll-f 1, \lj- 2, l | 11, ll 12, and I 15 by Hubert's method, and l | 9 by a newmethod (see §7). Hence it is clear that Ρ - L is not periodic.
§10. Periodicity of the Euler characteristic of an (M — 1)-variety
1. Recently Kharlamov in [129] and simultaneously and independentlyGudkov and Krakhnov in [128] have extended Rokhlin's results andmethods in [126] to (M - l)-varieties and (M - l)-pairs. Kharlamov'sarticle is somewhat more complete and uses a more general apparatus.Therefore, our account here follows, on the whole, Kharlamov's paper. Inthis subsection the theorems are proved for arbitrary non-singular algebraicvarieties (in the sense of §9.5). The concepts and notation of §§9.5 and9.6 are preserved.
THE (σ, χ) LEMMA. // V is a compact Ka'hler manifold of complexdimension 2k, then
(97) o(V) ^ (-l)"x(F)mod 4,
where o{V) is the signature and x{V) is the Euler characteristic of V.PROOF. As we know,
The topology of real projective algebraic varieties 59
( )P . 1 P,1
where hp'qis the dimension of the space of harmonic forms of type (p,q)on V (see [156*]). Since h?·" = hfl* and ff-Ί = h2k^1· 2k~p, for even k wehave
σ ( 7 ) - χ ( 7 ) = 4( Σ ^ i- i , 2i_ v ^ i - i , 2i-i _ ^ fe2-1·2^)
Σ2t<ft.
If k is odd, then
ΝΠ 2i~ 1 2 ?\
ΔΛ h ' ) .2i+2j-l<2fc
The lemma is applicable to the non-singular variety CA for even η and toΥ for odd n.
ARITHMETICAL LEMMA. Let G he a free abelian group of finite rank,and let Q: G X G -*• Ζ be an even symmetric bilinear form. We denote thesignature of Q by a{Q). If | det Q | = 2, then
(98) o{Q) = ± 1 mod 8.
PROOF. We choose a basis el, e2, . . . , er in G and form the matrixC - II 2(e,·, e}-) \\. Its determinant is even, hence, its rows are linearlydependent mod Z 2 . Consequently, we may assume that the basis is suchthat the first row and column of C are even. Then the determinant
e2) . . . Q (e2, er)
= 4m.;
Q(er, e2) ... Q(er, er)
is odd, with m £ Z, and the matrixmaetQ 1 0
10
0defines an even symmetric and unimodular form on G 0 Z. By a standardlemma (see [147*]) the signature of this form is divisible by 8. Hence(98) follows.
LEMMA ON ANNIHILATORS.1 Let G be a free Abelian group of finiterank, Q: G X G ^ Z a symmetric bilinear form with \ det Q | = \,and Εand F subgroups of G that annihilate each other relative to Q. Let Ε Π F = 0.Then
(99) | det QE I = I det QF \ = ord Gl{E © F).
Kharlamov first used this lemma in his paper and called it obvious. The idea of the proof was communi-cated to me by Rokhlin. The trivial case when Ε - G and F = 0 is, of course, not treated. The (σ, χ)-lemma is absent from the article [128] by Gudkov and Krakhnov. This is why [128] is incomplete, aspointed out above.
60 D. A. Gudkov
PROOF. G has a basis {ex, . . ., em, em+1, . . ., em+k} such that{c^, . . ., cmem) is a basis of E. It is easy to see that all thect = 1, 1 < / < m, that is, {ex, ,. . ., em) is a basis of E. Next, det Qg. =£ 0,because otherwise the columns of [1 Q(e,·), e;- II (/, / = 1, 2, . . . , m) arelinearly dependent and there is a ξ Φ 0, ξ €= £, orthogonal to £ relative toQ, which contradicts the condition Ε Π F = 0. Likewise, det Q p =£ 0.
Since Q is unimodular and symmetric, there exists a basis {dv . . ., dm+ft}dual to fo, . . ., em+ft} so that Q(e,·, e;·) = δ;)·. Then {dm+1, . . ., dm+h}is a basis of F. Consequently {ex, . . ., em·, dm + 1, . . ., dm + h} is a basis ofE® F, and hence £ Θ F is of finite index in G; this index is equal to themodulus of the determinant of the matrix of the expression of the basisof Ε θ F in terms of the basis of G {because G has a basis{/i. /a. · • ·. /m+fc} and there are integers au a2, . . . , am+k such thatOjfj, 1 < i < m + k, is a basis of £ 0 F). It is not hard to compute thatthe required determinant is equal to det QF if {e1( . . ., em+h}is taken fora basis of G.
THEOREM ON (M - 1 )-V AR IETIES. Let η be even. Then from
(100) dim H*{A\ Z2) = dim H*{CA; Z2) - 2
it follows that
(101) xU) = o(CA) ± 2 mod 16 x).
PROOF. We denote by G+ the subgroup of Hn(CA) consisting of theelements that are invariant under the automorphismconj*: Hn(CA) -> Hn(CA) and by G_ the subgroup consisting of the% £ Hn(CA) for which conj» £ = - £. Let Q be the linear form definedon G = Hn(CA) by the intersection index. It is not hard to see thatG+ Π G_ = 0 and that G+ and G_ annihilate each other relative to Q.By the lemma on annihilators, | det QG \ = \ det QG \ = ord G/G+© G_.We denote by Η the subgroup of invariant elements
conj*: Hn{CA\ Z 2 ) -*· Hn(CA; Z 2 ) . Siace G+@ G_ is the complete inverseimage of Η under the projection Hn(CA) -+ Hn(CA)/2Hn(CA) = Hn(CA; Z 2)(an element ξ of the complete inverse image of Η is the sum of£(£ + conj* 1) and ±(£ - conj* £)), Hn(CA)/G+ θ G. is isomorphic toHn(CA; Z2)/H. If we write down the Smith sequence for X = CA andr = conj, then Φ = A, and from (100) and (67) for the case under dis-cussion it follows that all the yr vanish except one, and for this onedim (Ker yr) = 1. Therefore, by properties 2° and 3° of the Smith sequence,Hn(CA; Z2)/H consists of at most two elements (two if yn Φ 0). Conse-quently, | det QG | = | det QG | < 2. We denote the signature of Q on G+
In [128] and [129] this theorem is proved for complete regular intersections. Here the general formula-tion is in accord with Rokhlin's note [130]. The proof is the same if one uses Rokhlin's first lemma,which is more general.
The topology of real projective algebraic varieties 61
and G_ by σ+ and a_. It is not hard to see that a{Q) = o+ + a_. Therefore,it follows from (71) that
(102) l(A) lK 1 a(CA)-2a+, if „ = 0 mod 4.
From the (σ, x)-lemma for CA we have (n even) a(CA) = (-l)"/ 2x(C4)mod 4.It follows from Poincare duality that χΟ4) Ξ (-l)"/2dim H*(A; Z 2 ) mod 4and x(C4) Ξ dim77*(G4; Z 2) mod 4. Therefore it follows from (100) that
(103) χ(Α) = a(C4) + 2 mod 4.
It remains to examine two possibilities.1) Hn(CA; Z 2) = H. From the proof of Rokhlin's first theorem (see §9)
it follows that χ(Α) = a(CA) mod 16, which contradicts (103). This casemust be rejected.
2) Hn(CA; Z2)/H consists of two elements. By the lemma on annihilators,I det QG | = | det QG | = 2. If η = 0 mod 4, by Rokhlin's first lemma Q_is even, and by the arithmetical lemma σ_ = ± 1 mod 8. Therefore, from(102) we have (101), which agrees with (103). If η = 2 mod 4, thensimilarly, Q+ is even and we also havs (101).
THEOREM ON (M - D-PAIRS.1 Let η be odd, m even, and A non-empty. Then from
(104) dim ff.U; Z2) + 2a_ = dim Hn (Y; Z2) - 2
it follows that
(105) 2χ(5_) = σ(Υ) ± 2 mod 16.
The proof is similar to that of the theorem on (M - 1 )-varieties.2. In this subsection we examine concrete examples on complete regular
intersections. The concepts and notation of §9.7 are assumed to hold.EXAMPLE 1. For complete regular intersections {Case A) the theorem
on (M - \)-varieties asserts that for even η from
(106) dim H. (A; Z2) = χ? (mu ..., ms)-2
it follows that
(107) χ μ ) = σ « Κ . .., ro,)±2 mod 16,
where v* and aq are defined by (80) and (83).S S
a) For a surface A of order mx in RPq we find that
(108) dim //* (A; Z2) = m* — km\ + 6 ^ — 2
implies the congruence(109) χ (Λ) = - ^ - ( 4 - T O * ) ± 2 mod 16.
The footnote on p. 60 applies here too.
62 D. A. Gudkov
For even m t this congruence yields
(110) %{A) = ± 2 mod 16,
which was announced by Kharlamov in [127]. For a quartic surface I hadstated (110) earlier as a conjecture.
EXAMPLE 2. For complete regular intersections (Case B) the theoremon (M — l}-pairs asserts that for odd n, even ms, and A non-empty
(111) dimH*{A; Z2) + 2a. = 6«(TOi, . . . , ms)-2
implies that
(112) 2χ (5_) == τ« (mu . . . , ms) ± 2 mod 16,
where 6f and τ§ are defined by (85) and (86).
a) Let q = 2 and s = 1 (plane curves). I recall that the order mi of Ais even. In this case, if (RP 2 , A) is an (M - l)-pair, that is, if A is an(M - l)-curve, then
(113) Ρ - L = (m.i/2)2±l mod 8.
For sextic curves in RP2 it follows from (51) and (113) that all the curveiswhich were proved in [120] not to exist do not exist here either; see §7of the present survey.
§11. Cubic and quartic surfaces in RP3
1. A rather vast body of literature is devoted to the investigation ofcubic surfaces (see [2], [4]-[6], [42], [65]-[67], [71], [103], [110]). Wedenote the coordinates of a point in RP3 by [x: y: z: u]. Let F3 = 0 bea real non-singular cubic surface in RP 3 . A complete topological classificationof such surfaces has been known for a long time. We denote by Ρ the com-ponent isotopic to RP 2 in R P 3 ; by Pk the component with k glued handleswithout knotting or linking; and by (P, 1) the surface consisting of Ρ andan ovaloid.
THEOREM ON CUBIC SURFACES. In RP 3 there are only five typesof non-singular cubic surfaces, namely, (P, 1), P, P 1 ; P2, and P3.
I sketch the idea of a proof. F 3 has an odd component (that is, onethat intersects a line in an odd number of real points). We take a pointa 0 on this odd component and choose a coordinate system [x: y: z: u] inRP 3 so that a0 has the coordinates [0: 0: 0: 1]. Then the equation of F3
takes the form
F3 = Q3(x, y, z) + 2Q2(x, y, z)u + Qx(x, y, z)w2 = 0,
where Qs(x, y, z) is a homogeneous polynomial of degree s with realcoefficients. In the plane u = 0 we can choose a coordinate system sothat Q,(x, y, ζ) Ξ Ζ. Through a0 and q[%\ η: £: 0] we draw a line L:
(114) χ = t\, y = t\], ζ = ίζ, u = μ.
The topology of real protective algebraic varieties
L intersects F3 in two points (in addition to a0), whose coordinates(t: μ) are expressed by the formula
63
t: μ = -02 (S. η, E)*Vgj(E, η. . ζ) ξ
6. η, ζ)
L is tangent to F 3 at (t: μ) if C4(£, τ?, ξ) = Q\ - Q3 ξ = 0, that is, if qbelongs to C4 in u = 0. It is easy to see that by a small change in thecoefficients of F3 we can arrange that Ql, Q2, and Q3 have no singularities,are pairwise non-tangent, and have no common points. Next, by a smallchange in the coefficient of f3 in Q3 we can arrange that C4 has nosingularities. It is not hard to see that C4 has real points, because in anyplane section of F3 through a° there is a tangent line from a0 to thissection. Therefore, C4 can have only one of the five types j-, 1, 2, 3 and4, where j- denotes one oval inside another, and k denotes k ovals outsideeach other. On the one hand, it can be shown that if Ck has one of thesetypes, then F3 is of type (P, 1), P, Plf P2, or P3, respectively. On theother hand, it can be shown that each of these cases can be obtained. Foiexample, we take a curve Q3 = 0 consisting of an odd branch intersectingthe line ξ = 0 in three real points and an oval. We also require thatQ3 ξ > 0 inside the oval of Q3. Next we choose Q2
Ξ ε-Η, whereΗ = 0 is a conic without real points, and ε is arbitrarily small. Then thezurve C4 = 0 consists of four ovals 0 l 5 0 2, 0 3, and 0 4. These ovals are theprojections of ovals 0Ί, 0'2, 0 3, and 0 4 onto F3, and at each point of theprojections the lines L from a0 touch the surface. We can cut F3 by threeof the latter ovals without violating the fact that F3 is connected. The lastoval cuts out a topological circle from F3. Therefore, F3 is of type P3.
We tabulate the Euler characteristics and the numbers dim Z/*(F; Z 2)for non-singular surfaces F3:
Table 6
Type of surface
Euler char, of F3
dim //* (F3; Z2)
(P, 1)
3
5
ρ
1
3
P i
5
Pi
—3
7
P 3
—5
9
2. In the formulation of his 16th problem Hubert posed, in particular,the problem of the topological classification of the non-singular quarticsurfaces F 4 in R P 3 (see [15], [119]. This problem is still not fully solved.In 1886-1913, Rohn in his papers [9j, [24], [25] investigated surfaces F4
by projecting from a singular point. Suppose that F4 has no singular points.It is not hard to see that in this case there exists a quartic surface Φ 4
64 D. A. Gudkov
with a unique singular point a0 (simple double) and that byan arbitrarily small change in the coefficients of Φ 4 one can obtain a sur-face isotopic to F4 in RP3, that is, of the same type. We choose a co-ordinate system [x: y: z: u] in RP3 so that a0 has the coordinates[0: 0: 0: 1]. Then the equation of Φ 4 has the form
Φ 4 = Q^x, y, z) + 2Q3(x, y, z)u + Q2(x, y, z)u2 = 0.
The line L through a0 and q[%: η: ξ: 0] (see (114)) intersects Φ 4 in twocoincident points at a0 and at two more points bx and b2 with the co-ordinates
„ ν -9 3 (E. η. Ώ ± VQl (Ε. η. ζ ) - & (Ε. η. ζ)·& (ξ, η, ζ)( ί · μ ) ΐ ' 2 ~ 9* (Ε. η, ζ) "
Χ is tangent at 6j = 6 2 if ^ belongs to the following sextic curve in theplane u = 0. C6(£, η, ζ) = Q\ - Qn'Qi = 0. Rohn examined in detail thecase when a0 is isolated, but only partially the case when a0 is a node ofΦ 4 . At that time the classification of non-singular curves C6 was not known.Nevertheless, Rohn gave a preliminary classification of the surfaces F 4 . Inparticular, he proved that such a surface cannot have more than 12components, and he proved the following important fact:
For a non-singular curve C6 to be the projection of a quartic surfacefrom a simple double point a0 it is necessary and sufficient that thereexists a conic Q2 that touches C6 at six points.
This theorem shows what a complicated problem it is to constructquartic surfaces.
In 1967-1969 Utkin [118], [120] improved Rohn's method, eliminatedmany errors in his proofs, and investigated exhaustively the case when a0
is a node of Φ 4 . This enabled him to prove that a quartic surface cannothave any types other than the fully determined (logically possible) 147types, of which none has more than 11 components. Next, he proved severaltheorems that facilitate the construction of Φ 4 and constructed many ofthe 147 logically possible surfaces.
Kharlamov [127] proved that F4 cannot have more than 10 components.This estimate is sharp: Rohn [24] constructed a surface F 4 consisting of10 ovaloids.
We introduce the following notation: we denote by A: a surface in RP 3
consisting of k ovaloids outside each other; by R° the component of thissurface homotopic to zero in RP 3 and isotopic to a sphere with ρ handleswithout knotting or linking; and by Rl
p the component isotopic to a one-sheet hyperboloid with ρ - 1 handles without knotting or linking. Finally,we denote by (i?°, k) the type of surface consisting of a component R°and k ovaloids outside each other in the exterior of R°; and by (R', k)the surface consisting of R'p and k ovaloids outside each other in one of
The topology of real protective algebraic varieties 65
the domains into which R'p divides RP 3
Of the 147 logically possible types of surfaces listed by Utkin we rejectthose that fail to satisfy at least one of the following restricting conditions:the Harnack-Thom theorem, according to which dim«(F 4; Z 2 ) < 24 (see§9, (88)); Rokhlin's first theorem, according to whichdim i/»(F 4 ; Z 2) = 24 implies that x(F 4 ) Ξ 0 mod 16 (see §9, (93)); thetheorem on (M — 1 )-varieties, according to which dim / / * ( F 4 ; Z 2) = 22implies that x(F 4 ) Ξ ± 2 mod 16 (see §10, (110)).
Taking all this into account, we can claim that a non-singular surfaceFq in RP3 cannot have types other than those in Tables 7 and 8 and six
Table 7
Table 8
{φ)
(R°A R,°o
10, (Rj,B)\(R2V),\(R°,6)./(R°,5), (Rg,4),\ \ /
9, (R?,7)\(R 2
0 ,6). (R?,5), (R°,4), (Rj.3),
8, (R°,6)\(R°,5). (R°,4), (R°.3),/(RU
5,2).
7, (RV,5)X(R°,4), ( R 3 ° , 3 ) , / R J , Z ) ,
6, (R?,4)\(R°,3)V
more that are not in these tables: (R\, R®, 1) - in one of the domainsinto which R\ divides the space there lie R° and the ovaloid outside eachother; (R\, i?,) - the same one without the ovaloid; 2R\ - two components
66 D. A. Gudkov
R\; (.2/??, 1) ~ two componentsRi and an ovaloid, all outside each other;2R° - two components R^ outside each other; j- - an ovaloid inside another.
Of the last six surfaces 2R\, j-, and 2R\ obviously exist. Of the surfaces inTable 7 Hubert [19] constructed anM-surface (R\o, 1) in 1909, Utkin [118],[120], [134] constructed an {M - l)-surface {R\, 9) in 1969, and recently[132] anM-surface (R\, 5). From these constructions it follows rather simplythat all the surfaces in Table 7 below the solid line exist. I make the followingconjecture:
CONJECTURE ON QUARTIC SURFACES NOT HOMOTOPIC TO ZEROIN RP 3 . All the quartic surfaces in Table 7 exist. In particular, the M-surface(R\,9) exists.
The situation concerning the surfaces in Table 8 is more complicated. In1972 I conjectured (see [127]) that a quartic surface homotopic to zeroin RP 3 , that is, a surface in Table 8 must satisfy the condition
dim H,(Ft; Z2) < 22.
Kharlamov [127] has announced a general theorem: a non-singular surfaceA of order m, that is homotopic to zero in RP 3 must satisfy thecondition
dim H,(A; Z2) < dim H*{CA; Z2) - 2,
that is, an Λί-surface cannot be homotopic to zero in RP 3 . The proof hasnot yet been published. Quite recently Kharlamov told me that he hasproved a general theorem from which it follows that for the surfaces inTable 8
dim # , ( 4 ; Z2) < 20.
Thus, surfaces in the first and second rows of Table 8 do not exist. Thesurface 10 was constructed by Rohn [24] in 1911. It follows easily fromthis construction that there exist surfaces of type k, where k < 9. Theremaining surfaces in Table 8 below the solid broken line were constructedby Utkin [118], [120] in 1969.
CONJECTURE ON QUARTIC SURFACES HOMOTOPIC TO ZERO INRP 3 . Only those surfaces that lie below the dashed broken line in Table 8exist.
The consideration of surfaces of orders ml < 4 naturally leads to thefollowing problem: either to give an example of a non-singular surface oforder ml < 5 having a component with a knotted handle, or to provethat this is impossible.
Thus, the problem of the topological classification of surfaces F4 inRP 3 (posed by Hubert) is also the source of new conjectures and problemssuch as the problem of the topological classification of sextic curves inRP 2 .
In addition to the papers mentioned so far, there are several other
The topology ofreal projective algebraic varieties 67
articles devoted to the study of quartic surfaces in R P 3 ; see, for example,[3], [93], [99].
§12. Conclusion
1. There are many surveys devoted to the papers on the topology ofreal algebraic varieties (see [43], [44], [46], [47], [59], [74], [76],[79], [86], [87], [95], [96], [100], [104], [105], [108], [109], [119]).In addition, there are papers devoted to individual problems of this subject(for example, [38], [48], [53], [55], [72], [78], [83], [94], [98],[107]) and not covered by this survey. I am sure that the list of the basicliterature is not complete.
2. I mention that the complete classification of non-singular curves ofsome order m does not solve the problem of the topological type of acurve defined by a specific equation of degree m. Therefore, it is interestingto find a computational algorithm to solve this problem I know of onlyone paper on this problem, namely, an article by Polotovskii [133], inwhich the author gives an algorithm for determining the topological typeof a rough plane curve of even order. It is assumed that the equation ofthe curve has real rational coefficients.
3. Apparently, the present stream of articles on the topology of realalgebraic varieties will not be interrupted for a long time, as in the past.Quite a number of connections between this branch of mathematics andothers have been discovered.
Very recently, Kharlamov [131] has generalized Petrovskii's theorem toa wide class of real algebraic varieties.
Let CA be a compact complex-analytic subvariety for the complex spaceCPq, where CA is invariant under complex conjugation conj: CPq -> CPq.The real part of CA is denoted by A and η = dim A, that is, η is thecomplex dimension of CA.
In addition, if η is odd, it is assumed that we are given a compactcomplex-analytic subvariety CB of CPq invariant under complex conjugationand a real homogeneous polynomial / of even degree m in(x0, * ! , . . . , xq) such that the hypersurface regularly intersects CB in CA.The real part of CB is denoted by B. It is assumed that A is non-empty,and the subvarieties defined in Β by / > 0 and / < 0 are denoted byB+ and B_.
Next, also for odd n, let Υ be a ramified double covering of CB withramification along CA, and aut: Υ -*• Υ a smooth non-identity automorphismof this covering.
The notation of Weil [157*] is used. We denote by SBr'(X) the vectorspace of cohomology classes of the complex-valued differential forms ofdegree r on a complex Kahler manifold X; by 3Ba>b (X) the space of bi-homogeneous cohomology classes of degree (a, b); by rf-b the dimension of
68 D. A. Gudkov
the latter space.CA, CB, and Υ are Kahler manifolds. For odd η the following additional
notation is introduced: 3β\ (Υ) is the subspace of &βτ (Υ) consisting ofthe form ω for which aut*co = ω; 3£T_ {Y) is the subspace of Sir (Y)consisting of the forms ω for which aut*co = - ω . Next we set
ma+b(Y)=sea·b(Υ) η mv*(Y), &ea-b(Y) = <%a'b (γ) η ssa-+b(Y),ha+ b (Y) = dim mai b (Y), Al" b (Y) = dim Sialh (Y).
In [131] the following propositions are proved:KHAR LA MOV'S FIRST THEOREM. If η is even, then
(115) \%(A)-i\^hn/2-n/z(CA)-l.
KHARLAMOV'S SECOND THEOREM. // η is odd; then
(116) | χ (Β_)-χ(Β+) | < ^ + 1 ) / 2 ' ( n + 1 ) / 2 (Y),
(117) | 2χ (Β.) - 1 | < f e ( n + 1 ) / 2 · ( n + 1 ) / 2 (F).
When Cyl and CB are complete regular intersections of the non-singularhypersurfaces (78) and (79), Kharlamov expresses the right-hand sides of(115)-(117) in the form of polynomials in q, mx, . . . , ms. He shows that1) for q = 2 and s - 1 (m, even) (117) yields (19), that is, Petrovskii'sfirst theorem; 2) for q = 4 and s = 1 (117) yields the correspondingestimate (37) of the first Petrovskii-Oleinik theorem; 3) for q - 3, 5 ands = 1 (115) yields the corresponding estimates (39) of the first Petrovskii-Oleinik theorem (also for odd mx). Kharlamov thinks that (115) and (117)always yield (39) (for any mx and q - 1 odd) and (37) (for even m1 andq — 1 odd); see §6.
Kharlamov also shows that (116) for q = 3 and 5 = 2 yields (48) in Oleinik'sfirst theorem (m2 even); see §6.
4. A current problem in the study of the topology of real algebraicvarieties is the problem whether the estimates in the Harnack-Thom andHarnack—Rokhlin theorems are sharp; see §9. There are grounds tobelieve that they are.
In the case of plane curves both these estimates are sharp. However, ingeneral it is apparently reasonable to begin the solution of this problem withthat of a certain problem for plane algebraic curves.
We introduce the concept of a model curve. Suppose that q circles areembedded smoothly and bilaterally in R F 2 . Suppose that their images inRP2 are pairwise disjoint. We call the curve so formed in RP 2 an evenmodel curve and denote it by A*. We call the components of A* ovals.The concepts of even and odd ovals are defined for A*. Suppose that thenumber of even ovals of A* is Ρ and the number of odd ovals is L. ThenΡ + L = q. The numbers P+, Po, P_, L+, Lo, and Lx are also defined forA* (see Arnol'd's theorem on nests, §8).
Suppose that q circles are embedded smoothly and bilaterally in RP 2 and
The topology of real projective algebraic varieties 69
that one circle is embedded smoothly and unilaterally. Suppose that alltheir images in RP 2 are pairwise disjoint. We call the curve so formed inRP 2 an odd model curve A*. We call the bilaterally embedded componentsovals and the unilaterally embedded component an odd branch. We denoteby Ρ and L the numbers of even and odd ovals of A*. Then Ρ + L = q.
The problem at hand is as follows. Suppose that for even m an evenmodel curve A* consisting of \(m — 1) (m — 2) + 1 ovals is given. We arerequired to find conditions {on the mutual disposition of these ovals) thatare necessary and sufficient for the existence of an M-curve A of order misotopic to A* in RP2.
The problem for Λί-curves of odd order m is formulated correspondingly.First of all, we note that any non-singular curve satisfies some trivial
restrictions.We call two nests Γ\ and Γ 2 of A distinct if they have no common
oval. An empty oval is a nest of weight 1.PROPOSITION 1 (TRIVIAL RESTRICTIONS ON THE INTERSECTION
WITH A STRAIGHT LINE). If a non-singular curve of order m has two
distinct nests Γι and Γ 2 of weights sx and s2, then
(118) si + s2 < mil.
The proof is obvious.We call / nests Γ,-, 1 < i < /, strongly distinct if for each of these Γ,
there exists an oval (among the ovals of these nests) lying outside the outer-most oval of Γ(.
PROPOSITION 2 (TRIVIAL RESTRICTIONS ON THE INTERSECTIONWITH A CONIC C2)- If a non-singular curve of order m has five stronglydistinct nests Γ,- of weights st, 1 < / < 5, then
s
(119) 2s ;<m.i=l
The proof is obvious.We say that the nests Γ,·, 1 < ι < /, lie outside each other if their
outermost ovals lie outside each other.PROPOSITION 3 (TRIVIAL RESTRICTIONS ON THE INTERSECTION
WITH A CURVE Cn). If a non-singular curve of order m has n(n + 3 ),
3 < η < m - 3, nests Γ;·, 1 < i < ^n + 3 ) , lying outside each other, then
there exist 3n - 1 of these nests Γ\, . . . , Γ 3 Λ _ 1 whose weightsSi, . . . , 5 3 n _ 1 satisfy the inequality
(120) ^ S i ^ n - n - { n " 2 ' ) { n - 2 ) .
For the proof one must keep in mind that there exists a curve Cn
passing through n^n + 3^ points. Cn may have \{n — 1) (n — 2) + 1 compon-
ents (but no more). It is possible that Proposition 3 can be replaced by a
70 D.A. Gudkov
stronger statement.If there exists an Λί-curve of even order m = 2k, then it satisfies the
following (necessary) conditions: 1) Ρ + L = \{m - 1) {m — 2) + 1; 2) thetrivial restriction on the intersection with a line; 3) the trivial restriction onthe intersection with a curve C2', 4) the trivial restriction on the inter-section with curves C r t , 3 < « < m - 3 ; 5 ) Petrovskii's first theorem, thatis, (19) in §6; 6) Ρ - L Ξ (m/2)2 mod 8; 7) Arnol'd's theorem on nests(see (57) and (58) in §8; in addition, it is highly probable that 8) Ragsdale'sconjecture holds, according to which Ρ < 3 m ' ~ 6 m + 1 and L < 3 m * " 6 m
8 8(a stronger proposition that Petrovskii's theorem).
Can one expect that the totality of l)-8) is sufficient, that is, that if aneven model curve A* satisfies these conditions, then there exists an Af-curveA of order m isotopic to A* in R ? 2 ? Apparently this is not the case.
We say that an oval a of A has weight s if a nest of maximal weight(of A) lying inside a has weight s. We denote by Us, the number of ovals of weight
s and we call U = ^ sU$ the weight of A. ForA/-curves of order m constructed
by the standard methods the following estimates hold:
) for m = Zk-l,
(121)
9
, 2 iifc-17 for m==2k, k odd, k > 3,2
gj. |0k2 5 — for m = 2k, k even, k > 2.
If (121) holds for any Λί-curve, then for even m > 10 this estimateyields additional restrictions on the disposition of the ovals in comparisonwith l)-8). For odd m > 9 (121) yields additional restrictions incomparison with 1)—4).
One can also introduce the concept of the coweight of a — the numberof ovals inside of which a lies. We denote the number of ovals of coweight
r by Vr, and we call V= ^r-Vr the coweight of A. Estimates for ther
coweight analogous to (121) can be observed experimentally. One can alsointroduce other quantities that characterize the mutual disposition of theovals of an M-curve.
Here are some more conjectures.CONJECTURE ON CURVES OF ORDER SEVEN. All 15 types of In-
curves of order seven listed at the end of §9 exist.CONJECTURE ON OCTIC CURVES. For octic M-curves 1), 2), 3), 6),
and 8) are sufficient, that is, for each of the 102 model curves A* satisfy-ing these conditions there exists an octic.M-curve A isotopic to A* inRP3.
CONJECTURE ON THE WEIGHT OF AN Λί-CURVE. For the weight ofan M-curve of order m (121) holds.
The topology of real protective algebraic varieties 71
Finally, one last remark. We have seen (§9) that if A is a curve in
RP 2 then ( R P 2 , A) is an Λί-pair if and only if A is an Λί-curve. The
examples of curves on a quadric in RP 3 show that the case when A is an
Λί-curve, but {B, A) is not an Λί-pair is possible. The case when A and Β
are Λί-varieties, but (B, A) is not an M-pair is also possible. The same
examples give reason to assume that the following theorem is probable:
// (B, A) is an M-pair, then both A and Β are an M-variety.
ADDED IN PROOF. Khailamov has told me that he has proved two new theorems supplementingthe material in § §9 and 10 of the present survey. I state these theorems in the terminology andnotation of § §9 and 10.
THEOREM I. If A is a complete regular intersection of hypersurfaces of orders m , , . . . ,mt in
KPQ with dim Η*(Α; Z 2) = x'(ml, . . . , ms) - 2 and if the inclusion homomorphism
Hnl2 {A; Z,) ->- IIn/4 ( R P 9 ; Z 2 ) is zero, then
l c." rn^m-i.. .ms ΞΞ 2 mod 4.
2 if m-i ... m. = 2 mod 8,
_ 2 if m ; . . . m : ^ -
T H E O R E M 2. Let A be a complete regular intersection of hypersurfaces of orders ml, . . . ,mgin
R P Q with d im Η*(Α; Z 2 ) = x ' ( m , . • • · > ms) - 4, a « d suppose that the inclusion homomorphism
is zero.
1°. // m, . . . m 4 = 0 mod 8, f/ien
χ (Λ) ΞΞ ± of (m,, . . . , ) H s ) m o d l 6 or χ (Λ) Ξ — σ« (m,, . . . , m s ) + 4 m o d l 6 .
2°. / / m , . . . m s = 2 mod 8,
χ (A) = ± σ^ («π, . . . , m s ) m o d l 6 or χ (.1) ΞΞ — σ^ (mj, . . . , m s ) + 4 m o d l 6 .
3". // m t . . . ms = - 2 mod 8, then
χ (A) = ± σ« (m ( m s ) mod 16 or χ (4) ΞΞ — σ^ («ij, . . . , m s ) — 4 mod 16.
It follows from Theorem 1, in particular, that there is no quartic surface A in R P 3 withdim H*(A; Z 2 ) = 22 that shrinks to a point.
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Received by the Editors, 5 November 1973
Translated by R. Lenet