Glossary of classical algebraic geometryWikipedia

The terminology of algebraic geometry changed drastically during the twentieth century, with the introduction ofthe general methods, initiated by David Hilbert and the Italian school of algebraic geometry in the beginning of thecentury, and later formalized by Andr Weil, Serre and Grothendieck. Much of the classical terminology, mainlybased on case study, was simply abandoned, with the result that books and papers written before this time can behard to read. This article lists some of this classical terminology, and describes some of the changes in conventions.Dolgachev (2012) translates many of the classical terms in algebraic geometry into scheme-theoretic terminology.Other books dening some of the classical terminology include Baker (1922), Coolidge (1931), Coxeter (1969),Hudson (1990), Salmon (1879), Semple & Roth (1949).Contents :

Conventions !$@ A B C D E F G H I J K L M N O P Q R S

1

2 1 CONVENTIONS

T U V W XYZ See also References

1 ConventionsOn the other hand, while most of the material treated in the book exists in classical treatises in algebraic geometry,their somewhat archaic terminology and what is by now completely forgotten background knowledge makes thesebooks useful to but a handful of experts in the classical literature.(Dolgachev 2012, p.iiiiv)

The change in terminology from around 1948 to 1960 is not the only diculty in understanding classical algebraicgeometry. There was also a lot of background knowledge and assumptions, much of which has now changed. Thissection lists some of these changes.

In classical algebraic geometry, adjectives were often used as nouns: for example, quartic could also be shortfor quartic curve or quartic surface.

In classical algebraic geometry, all curves, surfaces, varieties, and so on came with xed embeddings intoprojective space, whereas in scheme theory they are more often considered as abstract varieties. For example,a Veronese surface was not a just copy of the projective plane, but a copy of the projective plane together withan embedding into projective 5-space.

Varieties were often considered only up to birational isomorphism, whereas in scheme theory they are usuallyconsidered up to biregular isomorphism. (Semple & Roth 1949, p.2021)

Until circa 1950, many of the proofs in classical algebraic geometry were incomplete (or occasionally justwrong). In particular authors often did not bother to check degenerate cases.

Words (such as azygetic or bid) were sometimes formed fromLatin orGreek roots without further explanation,assuming that readers would use their classical education to gure out the meaning.

...we refer to a certain degree of informality of language, sacricing precision to brevity, ..., and which has longcharacterized most geometrical writing. ...[The meaning] depends always on the context and is invariably assumedto be capable of unambiguous interpretation by the reader.(Semple & Roth 1949, p.iii)

Denitions in classical algebraic geometry were often somewhat vague, and it is futile to try to nd the precisemeaning of some of the older terms because many of them never had a precise meaning. In practice this didnot matter much when the terms were only used to describe particular examples, as in these cases their meaningwas usually clear: for example, it was obvious what the 16 tropes of a Kummer surface were, even if tropewas not precisely dened in general.

3 Algebraic geometry was often implicitly done over the complex numbers (or sometimes the real numbers).

Readers were often assumed to know classical (or synthetic) projective geometry, and in particular to have athorough knowledge of conics, and authors would use terminology from this area without further explanation.

Several terms, such as Abelian group, complete, complex, at, harmonic, homology, monoid,normal, pole, regular, now have meanings that are unrelated to their original meanings. Other terms,such as circle, have their meanings tacitly changed to work in complex projective space; for example, a circlein complex algebraic geometry is a conic passing through the circular points at innity and has underlyingtopological space a 2-sphere rather than a 1-sphere.

Sometimes capital letters are tacitly understood to stand for points, and small letters for lines or curves.

2 Symbols[1], [2], . . . , [n] Projective space of dimension 1, 2, ..., n. This notation was introduced by Schubert (1886)

, , ... A family of dimension 1, 2, ...

{1}, {2}, ...,{n} A family or variety of dimension 1, 2, ..., n. (Semple & Roth 1949, p.288)

3 AAbelian group 1. An archaic name for the symplectic group.

2. A commutative group.

aberrancy The deviation of a curve from circular form. See Salmon (1879, p. 356)

absolute 1. A xed choice of something in projective space, used to construct some other geometry from projectivegeometry. For example, choosing a plane, called the absolute plane, of projective space can be used to makeits complement into a copy of ane space. Choosing a suitable conic or polarity, called the Cayley absolute,absolute conic or absolute polarity, in the absolute plane provides the means to put a metric on ane spaceso that it becomes a metric space.

2. Absolute geometry is roughly Euclidean geometry without the parallel postulate.

accidental An accidental (or improper) double point of a surface in 4-dimensional projective space is a double pointwith two distinct tangent planes. (Baker 1933, vol 6, p. 157)

acnode An acnode is an isolated point of a real curve. See Salmon (1879, p.23)

adjoint If C is a curve, an adjoint of C is a curve such that any point of C of multiplicity r has multiplicity at leastr1 on the adjoint. Sometimes the multiple points of C are required to be ordinary, and if theis condition isnot satised the term sub-adjoint is used. (Semple & Roth 1949, p.55, 231)

ane 1. Ane space is roughly a vector space where one has forgotten which point is the origin

2. An ane variety is a variety in ane space

anity An automorphism of ane space

aggregate A set.

ambient An ambient variety is a large variety containing all the points, curves, divisors, and so on that one is inter-ested in.

anharmonic ratio Cross-ratio

antipoint One of a pair of points constructed from two foci of a curve. See Salmon (1879, p.119)

4 4 B

apparent An apparent singularity is a singularity of a projection of a variety into a hyperplane. They are so calledbecause they appear to be singularities to an observer at the point being projected from. (Semple & Roth 1949,p.55, 231)

apolar Orthogonal under the polar pairing between the symmetric algebra of a vector space and its dualarithmetic genus The arithmetic genus of a variety is a variation of the Euler characteristic of the trivial line bundle;

see Hodge number.Aronhold set One of the 288 sets of 7 of the 28 bitangents of a quartic curve corresponding to the 7 odd theta

characteristics of a normal set.associated 1. An associated curve is the image of a projective curve in a Grassmannian, given by taking the tangent

lines, or osculating planes, and so on.axis, axial A special line or linear subspace associated with some family of geometric objects. For example, a special

linear complex in 4-dimensional space consists of all lines meeting a given plane, that is called the axial planeof the complex. (Semple & Roth 1949, p.274) Similar to directrix.

azygetic Unpaired. Opposite of syzygetic, meaning paired. Example: azygetic triad, azygetic tetrad, azygetic set.

4 Bbase 1. A base point is a point common to all members of a family2. The base number is the rank of the NeronSeveri group.bicircular Having nodes at the two circular points at innity, as in bicircular curve. See Salmon (1879, p.231)bicorn A bicorn is a curve with two cuspsbicuspidal Having two cuspsbidegree A pair of integers giving the degrees of a bihomogeneous polynomial in two sets of variablesbielliptic 1. A bielliptic curve is a branched double cover of an elliptic curve.2. A bielliptic surface is the same as a hyperelliptic surface.bid 1. Split into two equal parts2. A bid map is an element of the vector space of dimension 2g over the eld with 2 elements, consisting of the

2g+1-dimensional space of even-cardinality subsets of a set S of 2+2g elements, modulo the 1-dimensionalspace {0,S}. (Dolgachev 2012, p.215)

3. A bid substitution is a permutation of the 28 bitangents of a quartic curve depending on one of the 35decompositions of 8 symbols into two sets of 4 symbols. See Salmon (1879, p.223)

biecnode Same as eecnode. See (Salmon 1879, p.210)bigenus The second plurigenus P2 of a surface.bihomogeneous Homogeneous in each of two sets of variables, as in bihomogeneous form.binary Depending on two variables, as in binary formbinodal Having two nodesbinode A double point of a surface whose tangent cone consists of two dierent planes. See unode. (Semple & Roth

1949, p.424)bipartite Having two connected components. See Salmon (1879, p.165)bipunctual 1. Having two points2. For a bipunctual conic with respect to 3 points see (Baker 1922, vol 2, p. 123).

5birational Two varieties are birational if they are isomorphic o lower-dimensional subsets2. A birational map is a rational map with rational inversebiregular 1. A biregular map is a regular map with regular inverse2. Two varieties are biregular if there is a biregular map from one to the other, in other words if they are isomorphic

as abstract varieties.biscribed Both circumscribed and inscribed, or in other words having vertices that lie on a curve and sides that are

tangent to the curve, as in biscribed triangle. (Dolgachev 2012)bitangent A bitangent is a line that is tangent to a curve at two points. See Salmon (1879, p. 328)bitangential Meeting a curve at the tangency points of its bitangentsBrianchon hexagon A non-planar hexagon whose three diagonals meet. (Baker 1922, p.47)

5 Ccanonical 1. The canonical series is the linear series of the canonical line bundle2. The canonical bundle is the line bundle of dierential forms of highest degree.3. The canonical map or canonical embedding is the map to the projective space of the sections of the canonical

bundle4. A canonical curve (or variety) is the image of a curve (or variety) under the canonical map5. The canonical class is the divisor class of a canonical divisor6. A canonical divisor is a divisor of a section of the canonical line bundle.catalecticant A catalecticant is an invariant of a binary form of degree 2n that vanishes when the form is a sum of

powers of n linear forms.caustic A caustic is the envelope of light rays from a point reected in a curveCayleyCayleyan Named after Arthur Cayley1.

Main article: CayleyanSee Salmon (1879)

2. A Cayley octad is a set of 8 points in projective space given by the intersection of three quadrics. (Dolgachev2012, 6.3.1)

3. The Cayley lines or CayleySalmon lines are the 20 lines passing through 3 Kirkman points.4. A Cayley absolute is a conic or quadric used to dene a metric.centercentre 1. A special point associated with some geometric object2. The center of a perspectivity3. The center of an isologuecharactercharacteristic 1. An integer associated with a projective variety, such as its degree, rank, order, class, type. (Semple

&Roth 1949, p.189) In particular thePlcker characteristics of a curve are the order, class, number of nodes,number of bitangents, number of cusps, and number of inections. (Coolidge 1931, p.99)

6 5 C

2. A characteristic exponent is an exponent of a power series with non-negative coecient, that is not divisible bythe highest common factor of preceding exponents with non-zero coecients. (Coolidge 1931, p.220)

3. The characteristic series of a linear system of divisors on a surface is the linear system of 0-cycles on one of thedivisors given by its intersections with the other divisors.

chord A line joining two points of a varietychordal variety A chordal variety is the union of the chords and tangent spaces of a projective varietycircle A plane conic passing through the circular points at innity. For real projective geometry this is much the

same as a circle in the usual sense, but for complex projective geometry it is dierent: for example, cicles haveunderlying topological spaces given by a 2-sphere rather than a 1-sphere.

circuit A component of a real algebraic curve. A circuit is called even or odd depending on whether it has an evenor odd number of intersections with a generic line. (Coolidge 1931, p. 50)

circular 1. A circular point is one of the two points at innity (1: i: 0), (1: i: 0) through which all circles pass2. A circular algebraic curve is a curve passing through the two circular points at innity. See also bicircular.circumscribed 1. Having edges tangent to some curve, as in circumscribed quadrilateral.2. Passing through the vertices of something, as in circumscribed circle.cissoid A cissoid is the curve generated from two curves and a point. See Salmon (1879)class 1. The class of a plane curve is the number of proper tangents passing through a generic point of the plane.

(Semple & Roth 1949, p.28)2. The class of a space curve is the number of osculating planes passing through a generic point of space. (Semple

& Roth 1949, p.85)3. The class of a surface in rdimensional projective space is the number of tangent planes meeting a generic codi-

mension 2 subspace in a line. (Semple & Roth 1949, p.28)4. The degree of a contravariant or concomitant in the covariant variables.coaxalcoaxial A pencil of circles is called coaxal if their centers all lie on a line (called the axis).A family of plane circles all passing through the same two points (other than the circular points at innity). (Baker

1922, vol 2, p. 66)coincidence 1. A coincidence quadric is a quadric associated to a correlation, given by the locus of points lying in

the corresponding hyperplane. (Semple & Roth 1949, p.8)2. A xed point of a correspondence, in other words a point of a variety corresponding to itself under a correspon-

dence (Coolidge 1931, p. 126)collinear On the same linecollineation A collineation is an isomorphism from one projective space to another, often to itself (Semple & Roth

1949, p.6) See correlation.complete 1. A linear series of divisors is called complete if it is not contained in a larger linear series.(Semple &

Roth 1949, p.351)2. A scheme is called complete if the map to a point is proper3. A complete quadrangle is 4 points and the 6 lines joining pairs4. A complete quadrilateral is 4 lines meeting in pairs in 6 pointscomplex 1. (Noun.) A line complex, a family of lines of codimension 1 in the family of all lines in some projective

space, in particular a 3-dimensional family of lines in 3-dimensional projective space. (Semple & Roth 1949,p.236) See congruence.

72. (Adjective.) Related to the complex numbers.3. The (line) complex group is an old name for the symplectic group.composite Reducible (meaning having more than one irreducible component).conchoid A conchoid is the curve given by the cissoid of a circle and another curve. See Salmon (1879)concomitant A (mixed) concomitant is an invariant homogeneous polynomial in the coecients of a form, a covari-

ant variable, and a contravariant variable. In other words it is a (tri)homogeneous polynomial on SVVV*for some vector space V, where SV is some symmetric power of V and V* its dual, that is invariant under thespecial linear group of V. In practice V often has dimension 2. The degree, class, and order of a concomitantare its degrees in the three types of variable. Concomitants are generalizations of covariants, contravariants,and invariants.

concurrent Meeting at a pointcone 1. The union of the lines joining an algebraic set with a linear algebraic set. Called a point-cone, line-cone, ...

if the linear set is a point, line, ...(Semple & Roth 1949, p.18)2. A subset of a vector space closed under multiplication by scalars.conguration A conguration is a nite set of points and lines (and sometimes planes), generally with equal numbers

of points per line and equal numbers of lines per point.confocal Having the same focicongruence A family of lines in projective space such that there are a nonzero nite number of lines through a

generic point (Semple & Roth 1949, p.238, 288). See complex.conic A conic is a degree 2 curve. Short for conic section, the intersection of a cone with a plane.conjugate 1. A conjugate point is an acnode. (Salmon 1879, p.23)2. A conjugate point is a point lying on the hyperplane corresponding to another point under a polarity.3. A conjugate line is a line containing the point corresponding to another line under a polarity (or plane conic).

(Baker & 1922 vol2, p.26)4. For harmonic conjugate see harmonic.connex A correspondence between a projective space and its dual.consecutive Innitesimally near. For example, a tangent line to a curve is a line through two consecutive points of

the curve, and a focal point is the intersection of the normals of two consecutive points.contravariant 1. A bihomogeneous polynomial in dual variables of x, y, ... and the coecients of some homo-

geneous form in x, y,... that is invariant under some group of linear transformations. In other words it is abihomogeneous polynomial on SVV for some vector space V, where SV is some symmetric power of V andV* its dual, that is invariant under the special linear group of V. In practice V often has dimension at least3, because when it has dimension 2 these are more or less the same as covariants. The degree and class of acontravariant are its degrees in the two types of variable. Contravariants generalize invariants and are specialcases of concomitants, and are in some sense dual to covariants.

coplanar In the same planecorrelation An isomorphism from a projective space to the dual of a projective space, often to the dual of itself. A

correlation on the projective space of a vector space is essentially the same as a nonsingular bilinear form onthe vector space, up to multiplication by constants. (Semple & Roth 1949, p.7)

coresidual See Salmon (1879, p.131)correspondence A correspondence from X to Y is an algebraic subset of XYcosingular Having the same singularitiescouple An ordered pair

8 6 D

covariant 1. A bihomogeneous polynomial in x, y, ... and the coecients of some homogeneous form in x, y,... thatis invariant under some group of linear transformations. In other words it is a bihomogeneous polynomial onSVV* for some vector space V, where SV is some symmetric power of V and V* its dual, that is invariantunder the special linear group ofV. In practiceV often has dimension 2. The degree and order of a covariant areits degrees in the two types of variable. Covariants generalize invariants and are special cases of concomitants,and are in some sense dual to contravariants

2. The variety dened by a covariant. In particular the curve dened by the Hessian or Steinerian covariants of acurve are called covariant curves. (Coolidge 1931, p.151)

Cremona transformation A Cremona transformation is a birational map from a projective space to itself

cross-ratio The cross-ratio is an invariant of 4 points on a projective line.

crunode Crunode is an archaic term for a node, a double point with distinct tangent directions.

cubic Degree 3, especially a degree 3 projective variety

cubo-cubic A cubo-cubic transformation is a Cremona transformation such that the homaloids of the transformationand its inverse all have degree 3. Semple & Roth (1949, p.179)

curve A curve together with an embedding into projective space.

cusp A cusp is a singular point of a curve whose tangent cone is a line

cuspidal edge The locus of the focal points of a family of planes (Semple & Roth 1949, p.85, 87)

cyclide A cyclide is a quartic surface passing doubly through the absolute conic. (Semple & Roth 1949, p.141)

6 Ddecic

decimic 1. (Adjective) Degree 10

2. (Noun) A degree 10 projective variety

deciency 1. The deciency of a linear system is its codimension in the corresponding complete linear system.

2. The deciency D of a plane curve is an approximation to its genus, equal to the genus when all singular pointsare ordinary, given by (n1)(n2)/2 (a1)(a2)/2 (b1)(b2)/2 ..., where n is the degree of the curve anda. b, ... are the multiplicities of its singular points. (Semple & Roth 1949, p.30), (Salmon 1879, p. 28)

degree 1. The number of intersection points of a projective variety with a generic linear subspace of complementarydimension

2. The number of points of a divisor on a curve

Desargues The Desargues gure or conguration is a the conguration of 10 lines and 10 points in Desarguestheorem.

desmic system A desmic system is a conguration of three desmic tetrahedra.

developable 1. (Noun) A 1-dimensional family of planes in 3-dimensional projective space (Semple & Roth 1949,p.85).

2. (Noun) The envelope of the normals of a curve

3. (Noun) Short for a developable surface, one that can be unrolled to a plane

4. The tangent developable of a curve is the surface consisting of its tangent lines.

5. Flat, as in developable surface

9director The director circle of a conic is the locus of points where two orthogonal tangent lines to the conic meet.More generally the director conic of a conic in regard to two points is dened in a similar way. (Baker 1922,vol 2, p. 26)

directrix A straight line, or more generally a projective space, associated with some geometric conguration, suchas the directrix of a conic section or the directrix of a rational normal scroll

discriminant The invariant (on the vector space of forms of degree d in n variables) which vanishes exactly whenthe corresponding hypersurface in Pn-1 is singular.

double curve A 1-dimensional singularity, usually of a surface, of multiplicity 2double point 1. A 0-dimensional singularity of multiplicity 2, such as a node.One of the two points xed by an involution of a projective line. (Baker & 1922 vol 2, p.3)double six The Schli double six congurationduad A set of two pointsdual 1. The dual of a projective space is the set of hyperplanes, considered as another projective space2. The dual curve of a plane curve is the set of its tangent lines, considered as a curve in the dual projective plane.3. A dual number is a number of the form a+b where has square 0. Semple & Roth (1949, p.268)

7 EEckardt point An Eckardt point is a point of intersection of 3 lines on a cubic surface.eective An eective cycle or divisor is one with no negative coecientselation A collineation that xes all points on a line (called its axis) and all lines though a point on the axis (called its

center).eleven-point conic The eleven-point conic is a conic containing 11 special points associated to four points and a

line. (Baker 1922, vol 2, p. 49)embedded An embedded variety is one contained in a larger variety, sometimes called the ambient variety.enneaedro A set of 9 tritangent planes to a cubic surface containing the 27 lines.envelope A curve tangent to a family of curves. See Salmon (1879, p. 65)epitrochoid An epitrochoid is the curve traced by a point of a disc rolling along another disc. Salmon (1879)equianeequianity An equianity is an equiane transformation, meaning an ane transformation preserving area.equianharmonic 1. Four points whose cross ratio (or anharmonic ratio) is a cube root of 12. An equianharmonic cubic is a cubic curve with j-invariant 0equivalence In intersection theory, a positive-dimensional variety sometimes behaves formally as if it were a nite

number of points; this number is called its equivalence.evectant A contravariant dened by Sylvester depending on an invariant. See Salmon (1879, p. 184)evolute An evolute is the envelope of the normal lines of a plane curve. See Salmon (1879, p. 40)exceptional 1. Corresponding to something of lower dimension under a birational correspondence, as in exceptional

curve, exceptional divisor2. An exceptional curve on a surface is one that corresponds to a simple point on another surface under a birational

correspondence. It is called an exceptional curve of the rst kind if it is transformed into a point of the othersurface, and an exceptional curve of the second kind if it is transformed into a curve of the other surface.

10 9 G

8 Ffacultative A facultative point is one where a given function is positive. (Salmon 1885, p.243)

at 1. (Noun) A linear subspace of projective space, such as a point. line, plane, hyperplane.

2. (Adjective) Having curvature zero

3. (Adjective) For the term at in scheme theory see at module, at morphism.

ecnode A double point that is also a point of inexion of one branch. (Cayley 1852). (Salmon 1879, p.210)

eecnode A double point that is also a point of inexion of both branches. (Cayley 1852).

ex Short for point of inection

focal 1. A focal point, line, plane, ... is the intersection of several consecutive elements of a family of linearsubspaces. (Semple & Roth 1949, p. 85, 252)

2. A focal curve, surface and so on is the locus of the focal points of a family of linear subspaces. (Semple & Roth1949, p.252)

focus A focal point. See Salmon (1879, p. 116), (Semple & Roth 1949, p. 85,251)

foliate singularity See (Semple & Roth 1949, p.422)

form 1. A homogeneous polynomial in several variables. Same as quantic.

2. A dierential form.

free intersection An intersection point of two members of a family that is not a base point.

freedom Dimension, as in degrees of freedom. (Semple & Roth 1949, p.26).

fundamental This term seem to be ambiguous and poorly dened: Zariski states: I can nd no clear-cut denitionof a fundamental curve in the literature.

1. The fundamental set or fundamental locus of a birational correspondence appears to mean (roughly) either theset of points where it is not a bijection or the set of points where it is not dened.

2. A fundamental point, curve, or variety is a point, curve, or variety in the fundamental set of a birational corre-spondence.

9 Ggrd, rd A linear or algebraic system of divisors of dimension r and degree d on a curve. The letter g is used for

linear systems, and the letter is used for algebraic systems

generator One of the lines of a ruled surface (Semple & Roth 1949, p.204) or more generally an element of somefamily of linear spaces.

Most particularly we refer to the recurrent use of such adjectives as `general' or `generic', or such phrases as `ingeneral', whose meaning, wherever they are used, depends always on the context and is invariably assumed tobe capable of unambiguous interpretation by the reader.(Semple & Roth 1949, p.iii)

generic 1. Not having some special properties, which are usually not stated explicitly.

2. A generic point is one having coordinates that are algebraically independent over the base eld.

3. The generic point of a scheme.

11

genus 1. The dimension of the space of sections of the canonical bundle, as in the genus of a curve or the geometricgenus of a surface

2. arithmetic genus of a surface

3. plurigenus

geometric genus The geometric genus is the dimension of the space of holomorphic n-forms on an n-dimensionalnon-singular projective variety.

grade The grade of a linear system of divisors on an n-dimensional variety is the number of free intersection pointsof n generic divisors. In particular the grade of a linear series of divisors on a curve is now called the degreeand is the number of points in each divisor (Semple & Roth 1949, p.345), and the grade of a net of curves ona surface is the number of free intersections of two generic curves. (Semple & Roth 1949, p.45) (Semple &Roth 1949, p.159)

Grassmannian A Grassmannian is a variety parameterizing linear subspaces of projective space

group 1. A group or point-group is an archaic term for an eective divisor on a curve. This usage is particularlyconfusing, because some such divisors are called normal, with the result that there are normal sub-groupshaving nothing to do with the normal subgroups of group theory. (Coolidge 1931)

2. A group in the usual sense.

10 Hharmonic 1. Two pairs of points on a line are harmonic if their cross ratio is 1. The 4 points are called a harmonic

set, and the points of one pair are called harmonic conjugates with respect to the other pair.

2. A harmonic cubic is an elliptic curve with j-invariant 1728, given by a double cover of the projective line branchedat 4 points with cross ratio 1

3. Satisfying some analogue of the Laplace equation, as in harmonic form.

4. The harmonic polar line of an inection point of a cubic curve is the component of the polar conic other thanthe tangent line. (Dolgachev 2012, 3.1.2)

5. A harmonic net is a set of points on a line containing the harmonic conjugate of any point with respect to anyother two points. (Baker 1922, p.133)

6. For harmonically conjugate conics see (Baker 1922, vol 2, p.122).

Hesse

Hessian Named after Otto Hesse.

1. A Hessian matrix, or a variety associated with it. See Salmon (1879, p.55)

2. The Hessian line is a line associated to 3 points A, B, C, of a conic, containing the three points given by theintersections of the tangents at A, B, C with the lines BC, CA, AB.

3. The Hessian point is a point associated to three lines tangent to a conic, whose construction is dual to that of aHessian line

4. The Hessian pair or Hessian duad of three points on a projective line is the pair of points xed by the projectivetransformations of order 3 permuting the 3 points. More generally the Hessian pair is also dened in a similarway for triples of points of a rational curve, or triples of elements of a pencil.

5. The Hesse conguration is the conguration of inection points of a plane cubic.

6. The Hesse group is the group of automorphisms of the Hesse conguration, of order 216.

hexad A set of 6 points

12 11 I

homaloid An element of a homaloidal system, in particular the image of a hyperlpane under a Cremona transfor-mation.

homaloidal 1. A homaloidal linear system of divisors is a linear system of grade 1, such as the image of the linearsystem of hyperplanes of projective space under a Cremona transformation. (Semple & Roth 1949, p.45)(Coolidge 1931, p. 442) When the linear system has dimension 2 or 3 it is called a homaloidal net or homa-loidal web.

2. Homaloidal means similar to a at plane.

homographic 1. Having the same invariants. See Salmon (1879, p.232)

2. A homographic transformation is an automorphism of projective space, in other words an element of the projectivegeneral linear group. (Salmon 1879, p.283)

homography An axis of homography is a line associated to two related ranges of a conic. (Baker 1922, vol2, p.16)

homology 1. As in homology group

2. A collineation xing all lines through a point (the center) and all points through a line (the axis) not containingthe center. See elation. This terminology was introduced by Lie.

3. An automorphism of projective space with a hyperplane of xed points (called the axis). It is called a harmonichomology if it has order 2, in which case it has an isolated xed point called its center.

Hurwitz curve

Hurwitz surface A Hurwitz curve is a complex algebraic curve of genus g>0 with the maximum possible number84(g1) of automorphisms.

hyperbolism Essentially a blow-up of a curve at a point. See Salmon (1879, p.175)

hypercusp A singularity of a curve of some multiplicity r whose tangent cone is a single line meeting the curve withorder r+1. (Coolidge 1931, p. 18)

hyperelliptic A hyperelliptic curve is a curve with a degree 2 map to the projective line.

hyperex Same as point of undulation: a point of a curve where the tangent line has contact of order at least 4.

hyperosculating point A point where the tangent space meets with order higher than normal.

hyperplane A linear subspace of projective space of codimension 1. Same as prime.

11 Iindex of speciality The dimension of the rst cohomology group of the line bundle of a divisor D; often denoted by

i or i(D). Semple & Roth (1949, p.381)

innitely near point A point on a blow up of a variety

inection

inexion An inection is a point where the curvature vanishes, or in other words where the tangent line meets withorder at least 3. Dierential geometry uses the slightly stricter condition that the curvature changes sign at thepoint. See Salmon (1879, p. 32)

inpolar quadric See (Baker 1922, volume III, p. 52, 88)

inscribed 1. Having vertices on a curve, as in inscribed gure.

2. Tangent to some lines, as in inscribed circle.

integral An integral is (more or less) what is now called a closed dierential form, or sometimes the result ofintegrating such a form..

13

1. An integral of the rst kind is a holomorphic closed dierential form.

2. An integral of the second kind is a meromorphic closed dierential form with no residues.

3. An integral of the third kind is a meromorphic closed dierential form whose poles are all simple.

4. A simple integral is a closed 1-form, or the result of integrating a 1-form.

5. A double integral is a closed 2-form, or the result of integrating a 2-form.

invariant (Noun) A polynomial in the coecients of a homogeneous form, invariant under some group of lineartransformations. See also covariant, contravariant, concomitant.

inversion An inversion is a transformation of order 2 exchanging the inside and outside of a circle. See Salmon(1879, p.103)

involute An involute is a curve obtained by unrolling a string around a curve. See Salmon (1879, p. 278)

involution 1. A transformation whose square is the identity. Cremona transformations that are involutions includeBertini involutions, Geiser involutions, and De Jonquires involutions.

irregularity The irregularity of a surface is the dimension of the space of holomorphic 1-forms on a non-singularprojective surface; see Hodge number.

isologue Given a Cremoma transformation T, the isologue of a point p is the set of points x such that p, x, T(x) arecollinear. The point p is called the center of the isologue.

12 JJacobian 1. The Jacobian variety of a curve

2. A Jacobian curve; see below

Jacobian curve The locus of double points of curves of a net. (Semple & Roth 1949, p.115)

Jacobian set The set of free double points of a pencil of curves. (Semple & Roth 1949, p.119)

Jacobian system The linear system generated by Jacobian curves. (Semple & Roth 1949, p.117)

join The join of two linear spaces is the smallest linear space containing both of them.

13 Kkenotheme An intersection of n hypersurfaces in n-dimensional projective space. (Sylvester 1853, Glossary p. 543

548) Archaic.

keratoid Horn-like. A keratoid cusp is one whose two branches curve in opposite direction; see ramphoid cusp.Salmon (1879)

Kirkman point One of the 60 points lying on 3 of the Plcker lines associated with 6 points on a conic.

Klein surface A Klein icosahedral surface, a cubic surface

Kronecker index The intersection number of two curves on a surface

Kummer surface Main article: Kummer surfaceA quartic surface with 16 nodes

14 15 M

14 L

Laguerre net A net V of plane curves of some degree d such that the base locus of a generic pencil of V is the baselocus of V together with d1 collinear points (Dolgachev 2012, theorem 7.3.5) (Coolidge 1931, p. 423)

lemniscate A lemniscate is a curve resembling a gure 8. See Salmon (1879, p.42)

limaon A limaon is a curve traced by a point on a circle rolling around a similar circle. See Salmon (1879, p.43)

line A line in projective space; in other words a subvariety of degree 1 and dimension 1.

line coordinates Projective coordinates. See Salmon (1879, p. 7)

linear Degree 1

linear system A linear system of divisors, given by the zeros of elements of a vector space of sections of a linebundle

locus 1-A subset of projective space given by points satisfying some condition

15 M

manifold An algebraic manifold is a cycle of projective space, in other words a formal linear combination of irre-ducible subvarieties. Algebraic manifolds may have singularities, so their underlying topological spaces neednot be manifolds in the sense of dierential topology. Semple & Roth (1949, p.1415)

meet The meet of two sets is their intersection.

Mbius tetrads Main article: Mbius congurationTwo tetrads such that the plane containing any three points of one tetrad contains a point of the other. (Baker1922, p.62)

model 1. A variety whose points (or sometimes hyperplane sections) correspond to elements of some family. Similarto what is now called a parameter space or moduli space.

2. A model for a eld extension K of a eld k is a projective variety over k together with an isomorphism betweenK and its eld of rational functions.

modulus A function of algebraic varieties depending only on the isomorphism type; in other words, a function on amoduli space

Moebius tetrads See #Mbius tetrads

monoid A surface of degree n with a point of multiplicity n1. (Semple & Roth 1949, p.187)

monoidal transformation ACremona transformation of projective space generated by a family of monoids with thesame point of multiplicity n1. More generally a blow-up along a subvariety, called the center of the monoidaltransformation. (Semple & Roth 1949, p.187)

multiple A multiple point is a singular point (one with a non-regular local ring).

multiplicity The multiplicity of a point on a hypersurface is the degree of the rst non-vanishing coecient ofthe Taylor series at the point. More generally one can dene the multiplicity of any point of a variety as themultiplicity of its local ring. A point has multiplicity 1 if and only if it is non-singular.

15

16 NNronSeveri group The NronSeveri group is the group of divisors module numerical equivalence.

nest Two components (circuits) of a real algebraic curve are said to nest if one is inside the other. (Coolidge 1931)

net 1. A 2-dimensional linear system. See pencil and web. See also Laguerre net.

2. A harmonic net is a set of points on a line containing the harmonic conjugate of any point with respect to anyother two points. (Baker 1922, p.133)

Newton polygon Main article: Newton polygonThe convex hull of the points with coordinates given by the exponents of the terms of a polynomial.

nodal A nodal tangent to a singular point of a curve is one of the lines of its tangent cone. (Semple & Roth 1949,p.26)

node A singular point p of a hypersurface f = 0, usually with the determinant of the Hessian of f not zero at p.(Cayley 1852)

node cusp A singularity of a curve where a node and a cusp coincide at the same point. (Salmon 1879, p. 207)

normal 1. A subvariety of projective space is linearly normal if the linear system dening the embedding is complete;see rational normal curve

2. Orthogonal to the tangent space, such as a line orthogonal to the tangent space or the normal bundle.

3. A normal intersection is an intersection with the expected codimension (given a sum of codimensions). (SempleRoth, p.16)

4. Local rings are integrally closed; see normal scheme.

null-polarity A correlation given by a skew symmetric matrix. A null-polarity of the projective space of a vectorspace is essentially a non-degenerate skew-symmetric bilinear form, up to multiplication by scalars. See alsopolarity. (Semple & Roth 1949, p.9)

17 Ooctad A set of 8 points

octic 1. (Adjective) Degree 8

2. (Noun) A degree 8 projective variety

order 1. Now called degree of an algebraic variety: the number of intersection points with a generic linear subspaceof complementary dimension. (Semple & Roth 1949, p.15)

2. The order of a covariant or concomitant: its degree in the contravariant variables.

3. The order of a Cremona transformation is the order (degree) of its homaloids. (Semple & Roth 1949, p.46)

ordinary An ordinary point of multiplicity m of a curve is one with m distinct tangent lines.

oscnode A double point of a plane curve that is also a point of osculation; in other words the two branches meet toorder at least 3. (Cayley 1852)

osculate Kiss; to meet with high order. See Salmon (1879, p. 356)

osculating plane A tangent plane of a space curve having third order contact with it.

outpolar quadric See (Baker 1922, vol 2, p. 33, volume III, p. 52)

16 18 P

18 PPappus 1. Pappus of Alexandria.

2. The Pappus conguration is the conguration of 9 lines and 9 points that occurs in Pappuss hexagon theorem.

parabolic point A point of a variety that also lies in the Hessian.

parallel 1. Meeting at the line or plane at innity, as in parallel lines

2. A parallel curve is the envelope of a circle of xed radius moving along another curve. (Coolidge 1931, p.192)

partitivity The number of connected components of a real algebraic curve. See Salmon (1879, p.165)

Pascal Short for Pascal line, the line determined by 6 points of a conic in Pascals theorem

pedal The pedal curve of C with respect to a pedal point P is the locus of points X such that the line through Xorthogonal to PX is tangent to C. (Salmon 1879, p.96)

pencil A 1-dimensional linear system. See Lefschetz pencil.

pentad A set of 5 points

pentahedron A union of 5 planes, in particular the Sylvester pentahedron of a cubic surface.

period The integral of a dierential form over a submanifold

perspectivity An isomorphism between two projective lines (or ranges) of projective space such that the lines joiningeach point of one line to the corresponding point of the other line all pass through a xed point, called the centerof the perspectivity or the perspector.

perspector The center of a perspectivity

perspectrix The line in Desargues theorem on which the intersections of pairs of sides of two perspective triangleslie

pinch A pinch point is a singular point of a surface, where the two tangent planes of a point on a double curvecoincide in a double plane, called the pinch plane. (Semple & Roth 1949, p.175)

pippian Introduced by Cayley (1857). Now called the Cayleyan. See also quippian.

Plcker Main article: Julius Plcker

1. For Plcker characteristic see characteristic

2. A Plcker line is one of the 15 lines containing 4 of the 20 Steiner points associated to 6 points on a conic. ThePlcker lines meet in threes at the 60 Kirkman points. (Dolgachev 2012, p.124)

plurigenus Plural plurigenera

The dth plurigenus of a variety is the dimension of the space of sections of the dth power of the canonical linebundle.

point-star A family of lines with a common point

polar 1. (Adjective) Related by a polarity

2. The polar conic is the zero set of the quadratic form associated to a polarity, or equivalently the set of self-conjugate points of the polarity.

3. (Noun) The rst polar, second polar, and so on are varieties of degrees n1, n2, ... formed from a point and ahypersurface of degree n by polarizing the equation of the hypersurface.(Semple & Roth 1949, p.11)

4. A polar or polar line is the line corresponding to a point under a polarity of the projective plane.

17

polarity A correlation given by a symmetrical matrix, or a correlation of period 2. A polarity of the projective spaceof a vector space is essentially a non-degenerate symmetric bilinear form, up to multiplication by scalars. Seealso null-polarity. (Semple & Roth 1949, p.9)

pole 1. The point corresponding to a hyperplane under a polarity.

2. A singularity of a rational function.

poloconic

polocubic

poloquartic The poloconic (also called conic polar) of a line in the plane with respect to a cubic curve is the locusof points whose rst polar is tangent to the line. (Dolgachev 2012, p. 156157)

polygonal A polygonal (or k-gonal) curve is a curve together with a map (of degree k) to the projective line. Thedegree of the map is called the gonality of the curve. When the degree is 1, 2, or 3 the curve is called rational,hyperelliptic, or trigonal.

porism 1. A porism is a corollary, especially in geometry, as in Poncelets porism. The precise meaning seems tobe controversial.

2. An arrangement of geometrical gures (such as lines or circles) that are inscribed in one curve and circumscribedaround another, as in Poncelets porism or Steiners porism. There seems to be some confusion about whetherporism refers to the geometrical conguration or to the statement of the result.

poristic Having either no solutions or innitely many (Semple & Roth 1949, p.186). For example, Poncelets porismand Steiners porism imply that if there is one way to arrange lines or circles then there are innitely many ways.

postulated A postulated object (point, line, and so on) is an object in some larger space. For example, a point atinnity of projective space is a postulated point of ane space. (Baker 1922)

postulation The postulation of a variety for some family is the number of independent conditions needed to forcean elements of the family to contain the variety. (Semple & Roth 1949, p.440)

power of a point Laguerre dened the power of a point with respect to an algebraic curve of degree n to be theproduct of the distances from the point to the intersections with a circle through it, divided by the nth powerof the diameter. He showed that this is independent of the choice of circle through the point. (Coolidge 1931,p.176)

prime A hyperplane of projective space. (Semple & Roth 1949, p.1)

primal A projective hypersurface. (Semple & Roth 1949, p.10)

projectivity An isomorphism between two projective lines (or ranges). A projectivity is a product of at most threeperspectivities.

propinquity A number depending on two branches at a point, dened by Coolidge (1931, p. 224).

proximate For proximate points see (Zariski 1935, p.9)

pure All components are of the same dimension. Now called equidimensional. (Semple & Roth 1949, p.15)

19 Qquadratic transformation 1. A Cremona transformation of degree 2. A standard quadratic transformation is one

similar to the map taking each coordinate to its inverse

2. A monomial transformation with center a point, or in other words a blowup at a point.

quadric Degree 2, especially a degree 2 projective variety. Not to be confused with quantic or quartic.

quadrisecant A quadrisecant is a line meeting something in four points

18 20 R

quadro-cubic, quadro-quartic A quadro-cubic or quadro-quartic transformation is a Cremona transformation suchthat the homaloids of the transformation have degree 2 and those of its inverse have degree 3 or 4. (Semple &Roth 1949, p.180, 188)

quantic A homogeneous polynomial in several variables, now usually called a form. Not to be confused with quarticor quadric.

quarto-quartic A Quarto-quartic transformation is a Cremona transformation such that the homaloids of the trans-formation and its inverse all have degree 4. (Semple & Roth 1949, p.187)

quaternary Depending on four variables, as in quaternary form.quartic Degree 4, especially a degree 4 projective variety. Not to be confused with quantic or quadric.quintic Degree 5, especially a degree 5 projective varietyquippian A quippian is a degree 5 class 3 contravariant of a plane cubic introduced by Cayley (1857) and discussed

by Dolgachev (2012, p.157). See also pippian.quotient ring The quotient ring of a point (or more generally a subvariety) is what is now called its local ring, formed

by adding inverses to all functions that do not vanish identically on it.

20 Rramphoid Beak-like. A ramphoid cusp is one whose two branches curve in the same direction; see keratoid cusp.

Salmon (1879, p.46)rank 1. The rank of a projective curve is the number of tangents to the curve meeting a generic linear subspace of

codimension 2. (Semple & Roth 1949, p.84)2. The rank of a projective surface is the rank of a curve given by the intersection of the surface with a generic

hyperplane. (Semple & Roth 1949, p.193) See order, class, type.range 1. The set of all points on a line. (Coxeter 1969, p.242)2. A labeled or nite ordered set of points on a line.rational 1. Birational to projective space2. Dened over the rational numbers.ray A line, especially one in a family of linesregular 1. A regular surface is one whose irregularity is zero.2. Having no singularities; see regular local ring3. Symmetrical, as in regular polygon, regular polyhedron4. Dened everywhere, as in regular (birational) mapregulus One of the two pencils of lines on a product of two projective planes or a quadric surface.related Two ranges (labeled sets) of points on a line are called related if there is a projectivity taking one range to

the other.representative manifold A parameter space or moduli space for some family of varietiesresidual The residual intersection of two varieties consists of the non-obvious part of their intersection.resultant 1. The resultant of two polynomials, given by the determinant of the Sylvester matrix of two binary forms,

that vanishes if they have a common root.2. A Cremona transformation formed from n correlations of n-dimensional projective space.(Semple & Roth 1949,

p.180)reverse Inverse (of a function or birational map)ruled Covered by lines, as in ruled surface. See also scroll.

19

21 SSn Projective space of dimension n.

Salmon conic The Salmon conic of a pair of plane conics is the locus of points such that the pairs of tangents to thetwo conics are harmonically conjugate. (Dolgachev 2012, p. 119)

satellite 1. If a line meets a cubic curve in 3 points, the residual intersections of the tangents of these points withthe cubic all lie on a line, called the satellite line of the original line. See Salmon (1879, p. 127)

2. A certain plane curve of degree (n1)(n2) constructed from a plane curve of degree n and a generic point.(Coolidge, p. 159161)

3. For satellite points see (Zariski 1935, p.8). Possibly something to do with base points.

scroll A ruled surface with an embedding into projective space so that the lines of the ruled surface are also lines ofprojective space.

secant 1. A line intersecting a variety in 2 points, or more generally an n-dimensional projective space meeting avariety in n+1 points.

2. A secant variety is the union of the secants of a variety.

secundum An intersection of two primes (hyperplanes) in projective space. (Semple & Roth 1949, p.2)

Segre 1. Named after either Beniamino Segre or Corrado Segre

2. A Segre variety or Segre embedding is the product of two projective spaces, or an embedding of this into a largerprojective space.

3. The Segre cubic is a cubic hypersurface in 4-dimensional projective space.

self-conjugate

self-polar 1. Incident with its image under a polarity. In particular the self-conjugate points of a polarity form thepolar conic.

2. A self-conjugate (or self-polar) triangle (or triad) is a triangle such that each vertex corresponds to the oppositeedge under a polarity.

3. A self-conjugate tetrad is a set of 4 points such that the pole of each side lies on the opposite side. (Dolgachev2012, p.123)

septic

septimic 1. (Adjective) Degree 7

2. (Noun) A degree 7 projective variety

3. (Noun) A degree 7 form

sextactic point One of the 27 points of an elliptic curve of order dividing 6 but not 3. (Salmon 1879, p.132)

sextic Degree 6, especially a degree 6 projective variety

simple A simple point of a variety is a non-singular point. More generally a simple subvariety W of a variety V isone with a regular local ring, which means roughly that most points ofW are simple points of V.

singular Special in some way, including but not limited to the current sense of having a singularity

skew Intersecting in a set that is either empty or of the expected dimension. For example skew lines in projective3-space do not intersect, while skew planes in projective 4-space intersect in a point.

solid A 3-dimensional linear subspace of projective space, or in other words the 3-dimensional analogue of a point,line, or plane. (Semple & Roth 1949, p.4)

special divisor An eective divisor whose rst cohomology group (of the associated invertible sheaf) is non-zero

20 22 T

spinode A cusp. (Cayley 1852), Salmon (1879, p.23)

star A collection of lines (and sometimes planes and so on) with a common point, called the center of the star.(Baker 1922, p.109)

stationary point A cusp. See Salmon (1879, p.23)

Steiner

Steinerian 1. Named after Jakob Steiner

2. A Steinerian is the locus of the singular points of the polar quadrics of a hypersurface. Salmon (1879)

3. A Steiner surface is a certain embedding of the projective plane into projective 3-space.

4. a Steiner point is one of the 20 points lying on 3 of the Pascal lines associated with 6 points on a conic.

SteinerHessian One of Cayleys names for the Cayleyan. See Salmon (1879, p. 352)

surface An abstract surface together with an embedding into projective space.

superabundance of a divisor on a surface. The dimension of the rst cohomology group of the correspondingsheaf.

symmetroid The zeros of the determinant of a symmetric matrix of linear forms

syntheme A partition of a set of 6 elements into 3 pairs, or an element of the symmetric group on 6 points of cycleshape 222. (Dolgachev 2012)

system A family of algebraic sets in projective space; for example, a line system is a family of lines.

syzygetic Paired. Opposite of azygetic, meaning unpaired. Example: syzygetic triad, syzygetic tetrad, syzygetic set,syzygetic pencil.

syzygy 1. A point is in syzygy with some other points if it is in the linear subspace generated by them.(Baker 1922,vol I, p. 33) A syzygy is a linear relation between points in an ane space.

2. An algebraic relation between generators of a ring, especially a ring of invariants or covariants.

3. A linear relation between generators of a module, or more generally an element of the kernel of a homomorphismof modules.

4. A global syzygy is a resolution of a module or sheaf.

22 Ttacnode A tacnode is a point of a curve where two branches meet in the same direction. (Cayley 1852)

tacnode-cusp A singularity of a plane curve where a tacnode and a cusp are combined at the same point.(Salmon1879, p.207)

tact-invariant An invariant of two curves that vanishes if they touch each other. See Salmon (1879, p.76)

tangent cone A tangent cone is a cone dened by the non-zero terms of smallest degree in the Taylor series at apoint of a hypersurface.

tangential equation The tangential equation of a plane curve is an equation giving the condition for a line to betangent to the curve. In other words it is the equation of the dual curve. It is not the equation of a tangent to acurve.

ternary Depending on three variables, as in ternary form

tetrad A set of 4 points

tetragram Synonym for complete quadrilateral

21

tetrahedroid A tetrahedroid is a special kind of Kummer surface.tetrahedron A geometric conguration consisting of 4 points and the 6 lines joining pairs. This is similar to the

lines and innite edges of a polyhedral tetrahedron, but in algebraic geometry one sometimes does not includethe faces of the tetrahedron.

tetrastigm Synonym for complete quadranglethreefold 1. (Adjective) Three-dimensional2. (Noun) A 3-dimensional varietytorsal generator. A generator of a scroll (ruled surface) that meets its consecutive generator. See (Semple & Roth

1949, p.204)torse Developable surface.transvectant An invariant depending on two forms.transversal A line meeting several other lines. For example, 4 generic lines in projective 3-space have 2 transversals

meeting all of them.triad A set of 3 pointstricircular A tricircular curve is one that passes through the circular points at innity with order 3.tricuspidal Having three cuspstrigonal A trigonal curve is one with a degree three map to the projective line. See hyperelliptic.trihedral A set of 3 planes A Steiner trihedral is a set of three tritangent planes of a cubic surface whose intersection

point is not on the surface. (Semple & Roth 1949, p.152)trilinear coordinates Coordinates based on distance from sides of a triangle:Trilinear coordinates.trinodal Having three nodestripartite Having three connected components. Salmon (1879, p.165)trisecant A line meeting a variety in 3 points. See trisecant identity.tritangent Meeting something in 3 tangent points, such as a tritangent conic to a cubic curve or a tritangent plane

of a cubic surface.trope A trope is a singular (meaning special) tangent space.(Cayley 1869, p.202) The word is mostly used for a

tangent space of a Kummer surface touching it along a conic.twisted A twisted cubic is a degree 3 embedding of the projective line in projective 3-spacetotal A set of 5 partitions of a 6-element set into three pairs, such that no two elements of the total have a pair in

common. For example, {(12)(36)(45), (13)(24)(56), (14)(26)(35), (15)(23)(46), (16)(25)(34)} (Dolgachev2012)

type The type of a projective surface is the number of tangent planes meeting a generic linear subspace of codimen-sion 4. (Semple & Roth 1949, p.193)

23 Uundulation A point of undulation of a curve is where the tangent meets the curve to fourth order; also called a

hyperex. See inection point. (Salmon 1879, p.35, 211)unibranch Having only one branch at a point. For example, a cusp of a plane curve is unibranch, while a node is

not.unicursal A unicursal curve is one that is rational, in other words birational to the projective line. See Salmon (1879,

p. 29)

22 27 SEE ALSO

unipartite Connected. See Salmon (1879, p.165)

unirational 1. A correspondence is called unirational if it is generically injective, in other words a rational map.(Semple & Roth 1949, p.20)

2. A variety is called unirational if it is nitely covered by a rational variety.

united point A point in the intersection of the diagonal and a correspondence from a set to itself.

unode A double point of a surface whose tangent cone consists of one double plane. See binode.

24 Vvalence

valency The valence or valency of a correspondence T on a curve is a number k such that the divisors T(P)+kP areall linearly equivalent. A correspondence need not have a valency. Semple & Roth (1949, p.368)

Veronese surface Main article: Veronese surfaceAn embedding of the projective plane in 5-dimensional projective space

virtual An estimate for something that is often but not always correct, such as virtual genus, virtual dimension, andso on. If some number is given by the dimension of a space of sections of some sheaf, the correspondingvirtual number is sometimes given by the corresponding Euler characteristic, and equal to the dimension whenall higher cohomology groups vanish. See superabundance.

25 Wweb A 3-dimensional linear system. See net and pencil. (Semple & Roth 1949, p.160)

Weddle surface Main article: Weddle surfaceA quartic surface in projective space given by the locus of the vertex of a cone passing through 6 points ingeneral position.

Weierstrass point Main article: Weierstrass pointA point on a curve where the dimension of the space of rational functions whose only singularity is a pole ofsome order at the point is higher than normal.

Wirtinger sextic Main article: Wirtinger sexticA degree 4 genus 6 plane curve with nodes at the 6 points of a complete quadrangle.

26 XYZZeuthenSegre invariant The ZeuthenSegre invariant is essentially the Euler characteristic of a non-singular pro-

jective surface.

27 See also Glossary of algebraic geometry Glossary of arithmetic and Diophantine geometry Glossary of commutative algebra Glossary of dierential geometry and topology Glossary of invariant theory

23

Glossary of Riemannian and metric geometry Glossary of scheme theory List of complex and algebraic surfaces List of surfaces List of curves

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Zariski, Oscar (1935), Algebraic surfaces, Classics in Mathematics, Berlin, New York: Springer-Verlag, ISBN978-3-540-58658-6, MR 1336146

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