GEORGE JENN1NGS

T A N G E N T I A L C H O W F O R M S - R A N K 1

H Y P E R S U R F A C E S I N A G R A S S M A N N I A N

ABSTRACT. Let X " ~ pN be an n-dimensional projective variety, and N - n - 1 <~ k <~ N - 1. T h e

closure in the Grassmannian G(k + 1, N + 1) of the set of k-planes meeting the smooth locus of X nontransversally is a tangential Chow form (TCF) of X.

TCF's are generally hypersurfaces. We show that a hypersurface is a TCF iffits conormal form has rank ~< 1, and that a TCF is a hypersurface iffsome quadric in the second fundamental form of X h a s r a n k ~ > n + k + l - N .

TCF's were introduced by Green and Morrison [3], generalizing a construc- tion of Cayley [1] and Chow and Van der Waerden [2]. Green and Morrison give local criteria for a hypersurface to be a TCF, and for a TCF to be a Chow form (CF). The CF is always a hypersurface. I fF is its defining polynomial then invariants of the coefficients of F furnish coordinates for an embedding of the

moduli space of X into a projective space-see [2], [5, pp. 31-32]. We work over C. G = G(m + 1, N + 1) is the Grassmannian of m planes in

Ps; equivalently, (m + 1)-dimensional subspaces of C ~+ 1.0--* S ~ C N+ 1

Q ~ Q is the universal vector bundle sequence over G. If A e G, then S A is just the subspace of C N÷ 1 representing A. There is a well-known canonical identification of the tangent bundle

(1) TG ~- Hom(S, Q), v ~ A v.

If e : q / c G ~ S is a section on a neighborhood of A and veT^G, then A~ o e(A) = ~e/av mod S A (differentiate e as a map into C N+ 1). Similarly,

(2) T* G ~ Horn(Q, S).

In particular, it makes sense to speak of the rank of a tangent vector (covector) on G.

SECOND F U N D A M E N T A L FORM (see Griffiths and Harris [4]). Let

X" c pn+k be an n-dimensional subvariety, X o c X the smooth locus. The

projective second fundamental form is a map

2 H: Symm T X o ~ N X o,

where N X o is the normal bundle of X o in pn+k. I f p ~. X 0 and (x 1 . . . . . x,;

Yl , . . . , Yk) is an 'affine' coordinate system such that dyl , . . . , dy k ~ N * X o, then

Geometriae Dedicata 28 (1988), 1-6. © 1988 by Kluwer Academic Publishers.

2 G E O R G E J E N N I N G S

(x 1 . . . . . x,) restricts to a coordinate system (x ' l , . . . , x~) on X o near p, and

H ' ~xi ~ l c~x~ dx; mod T~, X

at p. The associated quadrics are those in the linear system

2 IHI = imH*, H*: N * X o --, Symm T * X o,

2 '9 :2 Namely, if Q = H* dy~ then Q(O/Ox~) = 3 Yd xi • Consider

X = X n ~ pn+k an n-dimensional complex projective variety

X o is its smooth locus, and

G = G(r + k, n + k + 1) the Grassmann variety o f t + k - I planes in pn+k, where 0 <<. r <<. n.

(3) DEFINITION. The rth tanoential Chow form of X, TCF, X, is the closure in G of the locus of r + k - 1 planes meeting X o nontransversaUy. TCFoX is the Chow form of X and TCF, X = X "/, the dual of X.

Let TCF, X be the closure in X x G x p~+k,/ of the incidence variety {(p,A,H)~Xo x G x Pn+k'/IpeA , TpX u A = H}, where TpX c p~+k is the embedded tangent pla,~ne and p*+kV is the dual projective space. Let rcG: T ~ , X --, G, 7ix: TCF, X ---, X, and rope: TC"~, X ---, p,+k,/be the induced projections. Then

TCF, X = n~(TCF, X).

At a generic point (p, A, H) e TC--'~, X, ~ X c~ A ~- P~ and span(~ X, A) = H.

Thus, for generic p ~ X, the restriction of n~,/to the fiber nx l(p) is generically finite and dim 7~ x 1 (p) = dim tr, where tr = {A e G Ip e A and dim 7",X n A 1> r}. Hence

dim TCF, X = dim X + dim a = dim G - 1.

(4) THEOREM. TCF, X has codimension >1 2 in G if and only if every

quadric in the second fundamental form of X has rank < r.

(5) COROLLARY. The Chow form TCF o X is always a hypersurface.

Proof of the theorem. Let PoeXo. Given a generic ~o = (po,Ao, Ho)e 7tx l(po) there exists a neighborhood q / c nx ~(Xo) of Ao such that

(6) TpX c~ A ~ P,, span(~X,A) = H

for all A = (p, A, H) e q/. In particular, there exists a basis (eo . . . . . e,+k) for C ~+k+ ~ such that, projectivizing,

T A N G E N T I A L C H O W F O R M S 3

(7) P(span eo) = Po

0Z(span(eo... en)) = Tpo X

P(span(eo... e,, e. + 1... e, + k- 1)) = Ao

P(span(eo-.. e ,+k- 1)) = Ho"

Let [X o . . . . , X , , Y , + I , . . . , Y.+k] be the homogeneous coordinate system dual to this basis, and x i = X i / X o , i ~ O, yj = Y j X o, the associated affine coordinate system with origin at Po. Set

x} = xilx, y)(x], . . . ,x~) = YjIx, i = 1 . . . . . n,

j = n + l . . . . . n + k .

Then (x] . . . . , x'.) is a local coordinate system on a neighborhood ~ c X o of

Po, and

(8) ~YJ(po)=0, i = l , . . . , n , j = n + l , . . . , n + k .

Consider the bundle 7"X ~ ~//, where TpX is the plane representing TpX in C.+k+ 1. The sections

~, , n+k (9) eo(X' ) = e o + eixl + ~, e jy) (x ' )

i = l j = n + l

.+k Oy) ei(x') = e i + ~ eJox}, i = l . . . . . n,

j = n + l

determine a frame for ~ X over ~¢r with

(10) P(spaneo(x ' (p) ) ) = p, V p ~ t ¢ ".

Pull everything back to 0//~ T C F , X . After perhaps replacing q/ with a smaller neighborhood of/~o, it follows from (6-10) that there exist unique functions on ~'

~ j = r + 1, . . . ,n, A ~ ' B ~ ' C ' ( a = 1 , . . . , r , s = n + 1 . . . . . n + k - 1,

such that the sections

(11) E o = eo(x' )

E , = e,(x ') +

Es= j = r + l

, j ej,xt ~A,._a, a = l . . . . ,r, j = r + l

, j e j ( x ) B ~ + e ~ + e . + k C ~ , s = n + l . . . . . n + k - 1 ,

4 G E O R G E J E N N I N G S

of C" +k + 1 form a frame for the pullback rt~ S of the universal subbundle to q/.

(In other words, A = P(span(Eo E l . . . Er; E. + 1.., E. +k- 1 )) for all (p, A, H) E q/ - compare (7).)

It follows from (7)-(1 i) that (x', A, B, C) form a coordinate system on q/with origin at/~o. Using (1),

d~o()~o) = ~ e j ® e * ® d x j + ~ ~ e j®e*®dA~ j = r + l a = l j = r + l

n n + k - 1 n + k - 1

+ y~ e~®e*®de~+ ~ e .+k®e*®dC, j = J ' + 1 s = n + l s = n + l

, - f ~ y ' . + ~ ' X . + a=, ~ e"+k®e'cl~--~-x'.) m°aSA°'

where e*(ej) = 6 o, i,j = 0 .. . . . n + k. Since

d dy.+k = 0 y.+kdx,b mod {dx~Lj = r + 1, n} b= 1 dX'a ~X'b " ' "

it follows that

(12) rank dng(Ao) = dim G - 1 - r + q ,

2 ¢ t t where q = rank (0 y. + k/Ox~ dXb)a,b = 1 ...... . Clearly, if every Q ~ I HI has rank < r, then dno(Ao) has rank < dim G - 1. Conversely, if some Q ~ 1171 has rank >i r, then one can choose A o and (e o . . . . , e. + k) SO that Q = H* dy. +k and the matrix (12) has rank r. This completes the proof.

Fix a p . - , in p,+k. Let al, o = {A~ G(r + k,n + k + I)IA meets p . - , } and

g = dim c G. Then H2g_2(G, 7/) = Z.[a l ,o] . If TCFrX is a hypersurface then [TCFrX] = dr" [tr~,o] in H2g_2(G,Z) for some dr e Z, the degree of TCFrX.

(13) PROPOSITION. Assume X is irreducible. I fTCFrX is a hypersurface, then its degree is equal to the degree of the dual of a generic section of X by a pr + k in pn+k. In particular d o = degX (see also [2]).

Proof. By the Schubert calculus, d r is the intersection number dr = # [TCF~X] • [aT, o] with tr*,o = {A~ G I[P r+k- 2 ~ A c p,+k}, where pr+k-2 pr+k are fixed planes in p,+k. Set V~ = X c~ pr+k, SO r = dim Vr, and

= {A ~ TCFrXIA = pr+k}. Vr may be dualized in p . +k,/or in pr +k,/_ the former dual is just a cone over

the latter with vertex pr+k-. Their degrees are the same. Take the dual of V, to be its dual in 0 zr+k.

If pr+k is in sufficiently general position, then X o c~ p,+k is dense in V, and Xo ' pr+k meet transversally. Then V~ 4 ~ 7 since y is closed. Because

TANGENTIAL CHOW FORMS 5

rcx ~r ~ 1 (7) = V, it follows that V f = 7. Hence deg Vf = # { 2 e 7 ] P" + k- 2 c A } =

# [TCF, X] • [a*.o].

(14) THEOREM. A hypersurface Y ~ G is a tangential Chow form if and

only the generic form in its conormal sheaf has rank = 1.

Proof Let Y0 be the smooth locus of Y, Ao ~ Yo. (=*-) Assume Y = TC'~r X. Let ~o s rc~ ~(A0). The set of regular values of rc~ is dense in Yo, so without loss of generality one may assume that/~o is a smooth point of ~'C~I6,x and dTr~: TAo TCF, X ~ TAo TCF, X is surjectlve.

There exist a neighborhood q / c 7r~ t(Yo) of 7(o and nonvanishing sections e:q/--* rr*S and e*: a//--* rr*Q* such that for all A = (p,A,H)aq/

(15) P(span e(~)) = p and P(span e*(~)) = H.

Since N * Y o is a line bundle it suffices to show that e ® e* is a section of rc}(N* Yo) over q/.

Let v s TAoTCF, X , w = drco(v). Then at/~0,

~e . AwJ e ® e* = (A w • e)_le* = ~ . l e ,

where ] denotes contraction. But by (15), Oe/Ov =-0 rood ~I'pX, where ~ ' p X C C N+I is the plane representing TpX in pN. Since ~i 'pXCH,

~?e/OvJe* = O.

(~) Conversely, assume every form in N* Yo has rank ~< 1. Let ~ c I1o be an open neighborhood equipped with a nonvanishing

section e ® e*: ~K --, N* II0. After perhaps shrinking ~ one may decompose e ® e* into nonvanishing sections e: ~¢r ~ S and e*: ~/C --; Q*. Define maps p: ~K ~ pN and H: ~/r _., pN* by

( 1 6 ) P(span e) = p, P(span e*) = H.

If Aeq¢/" and V e T A Y o then A v J e ® e * = O , hence Oe/OvJe*=O, hence dp(v) - 0 rood Tp H where p = p(A), and H = H(A).

It follows that the image of p completes to a proper subvariety X of pN, SO Y c TCFrX for some r. Since Y is a hypersurface it consists of a union of irreducible components of TCF, X; since a TCF of an irreducible variety is irreducible one may, by throwing away superflous components of X, make Y = TCF~X.

(17) COROLLARY. In (14), Y c G(m + 1, N + 1)is a Chow form if and only if it is a TCF and the map (16) dp:T A Yo -o Tp pN has rank ~ N - m - 1 at every point A ~ Yo.

6 GEORGE JENNINGS

REFERENCES

1. Cayley, A., 'On a New Analytical Representation of Curves in Space', Quart. J. Pure Appl. Math. 3 (1860), 225-236; 5 (1862), 81-86.

2. Chow, W. L. and van der Waerden, B. L., 'Zur algebraischen Geometrie. IX. Ueber Zugeordnete Formen und algebraisehe Systeme yon algebraischen Mannigfaltigkeiten', Math. Ann. 113 (1936), 692-704.

3. Green, M. and M orrison, I., 'The Equations Defining Chow Varieties'. Duke Math. J. 53 (1986), 733-747.

4. GriffithS, P. and Harris, J., 'Algebraic Geometry and Local Differential Geometry', Ann. Scient. Ec. Norm. Sup., 4e s6rie, 12 (1979), 355-432.

5. Mumford, D., Curves and their Jacobians, Univ. of Michigan Press, 1975.

Author 's address:

George Jennings,

D e p a r t m e n t of Mathemat ics ,

Cal i forn ia Sta te Univers i ty ,

Dominguez Hills,

Carson, C A 90747,

U.S.A.

(Received, February 12, 1986)