Intersections of projective varieties and generic projections

manuscripta math. 92,273 - 286 (1997) n a n u s c r £ p t a m a t h e m a t J . c a g: Sprin~r-Yedag 199"/

Intersections of Projective Varieties and Generic Projections

Hubert Flenner and Mirella Manaresi *

Received August 18, 1995; in revised form November 20, 1996

Let X, Y C P~ be closed subvarieties of dimensions n and m respectively. Proving a Bezout theorem for improper intersections Stiickrad and Vogel [SVo] introduced cycles vk "= vk(X, Y) of dimension k on X N Y and/~k on the ruled join variety J := J(X, Y) of X and Y which are obtained by a simple algorithm..In this paper we give an interpretation of these cycles in terms of generic projections Pk : pN ~ pn+m-k-l. For this we introduce a relative ramification locus R(Pk, X, Y) of Pk which is of dimension at most k and generalizes the usual ramification cycle in the case X = Y. We prove that this cycle is just Vk for 0 < k < d imXCIY- 1. Moreover, the cycles flk+l (for - 1 < k < d i m X C t Y - 1) may be interpreted geometrically as the cycle of double points of Pk associated to the closure of the set of all (x : y) in the ruled join J such that (pk(x) : Pk(Y)) is in the diagonal A~,+~_k_: of j (pn+m-k-1, p'n+m-k-1).

1 I n t r o d u c t i o n

Let X, Y be closed subvarieties of the projective space pN = p~ where K is an arbi trary field. Proving a Bezout theorem for improper intersections Stiickrad and Vogel [SVo] introduced cycles vk = vk(X, Y) of dimension k on X Ct Y which are obtained by a simple algorithm on the ruled join variety

J : = J ( X , Y ) : = { ( x : y ) e P 2 n + ~ : x E X , y E Y }

in the following way. Let A := {(x : x) E p2N+l : x e pN} be the diagonal, so tha t A is given by the equations

X o - Y o . . . . . X ~ - Y ~ = 0 ,

where X 0 , . . . , XN, Y0 . . . . , Y~, are homogeneous coordinates in p2N+l. For indeterminates Uij (0 < i , j < N) let L be the pure transcendental extension K(Uq)o<i~i<N and let Hi c JL := J ®t¢L be the divisor given by the equation

t~ := z u i j ( x j - ~ ) = o .

Then one defines cycles flk and vk inductively by setting

/~dim Z := [J].

If flk is already defined, decompose the intersection

~kNHdimd_k='Ok_l-I-[Jk_l (1 < k < dim J ) ,

*This paper was supported by the SFB 170 "Geometry and analysis" at the University of G6ttingen. The second named author would like to thank this institution for its hospitality

274 Flenner-Manaresi

where the support of vk-1 lies in A and where no component of P~-l is contained in A. It follows that v~ is a k-cycle on XL n YL - JL n A L. In general, v = ~ ve is a cycle defined over L. By a result of van Gastel [vGa] a K-rational subvariety C of XL n YL occurs in v iff C is a distinguished subvariety of the intersection of X and Y (in the sense of Fulton [Ful]), and this is equivalent to the maximality of the analytic spread of the ideal of X N Y in Oj, c (see lAMa]) or the maximality of the dimension of the so called limit of join variety, see [FVo].

In this paper we will give a geometric interpretation also of the non K-rational components of v at least in the case when X, Y are smooth and meet smoothly (see 4.1).

More precisely, for a pair of closed subvarieties X, Y C ~N of dimension n and m, respectively, the sheaf fl~uv ® OxnY is locally free on the set Sm(X, Y) where X, Y and X n Y are smooth. Let p~ : X U Y , p~+m-k-I be the generic linear projection given by Pk := (A0 : " " : A .+m-k- J where Ai := ~ UijXj. We introduce the relative ramification set R(pk) to be the set of points x E Sm(X, Y) where the tangent map

TzPk : TXLUVL,~ --~ Tp~+--k-h~

has not maximal rank. Then the main result of section 4 is that the dimension of R(Pk) is at most k and that on Sm(X, Y) the k-cycle [R(pk)] is just vk, for O < k < d i m X N Y - 1 .

Also the cycle/~k+l (for - 1 < k < d i m X n Y - 1) admits a geometric description as the double point cycle of p~ which we introduce as the (k + 1)-cycle [D(pk)] associated to the closure of

{(~: 9) e ~'~: (pkCz): p~Cy)) e Ap~} \ AxL~

in JL where Ap L _C j (p~+m-k- l , p~+.~-~-l) denotes the diagonal subspace, see (3.8). Moreover we can prove that

codirr~(SingD \ A) > codimj(Sing J \ A).

We remark that for the case X = Y this slightly unusual definition of the double point cycle is essentially equivalent to the classical one, see (3.7).

In the case of selfintersection of a submanifold X ~-* pN these results can be described as follows. The set SIn(X, X) is just X, and the ramification cycle just the cycle associated to the set of ramification points R(pk) of the generic linear projection Pk for 0 < k < n - 1. I t follows that in this range vk is given by the k-dimensional cycle of ramification points of Pk. Moreover the only K-ra t ional component of v is [X] itself. From the description of/~k+l we have in particular that

deg flo = 26,

where 6 is the number of double points of p(X) under the generic linear projection p : X --- p2.. By a result of van Gastel [vGa]

deg /~o = deg(Sec X ) . deg(J/Sec X)),

where Sec X is the secant variety and d e g ( J / S e c X) is the degree of the rational map J - - ~ Sec X given by i x : y) ~-* (x - Y)- Thus we obtain the equality

26 = deg(Sec X ) . deg(J/Sec X) .

Flenner-Manaresi 275

Using the Stiickrad Vogel algorithm it follows in particular that

H.D(pk+I) = R(pk) ÷ D(pk),

where H is a hyperplane on J. This type of relation between the double point cycle and the ramification cycle was found by Johnson [Joh], see also [Ful], 9.3.13. and Hansen [Han]. Thus our interpretation of the Stiickrad Vogel cycles f~k, vk gives another geometric proof of the above identity.

We remark that for arbitrary X and Y van Gastel obtained a characterization of vk in terms of special Schubert classes on a normal cone, see [vGa]. However, the interpretation given here seems to be more natural and relies on direct geometric arguments which allow us to deduce further information about geometric and algebraic properties of the double point cycle and ramification cycle.

The paper is organized as follows. Section 2 is devoted to residual intersections. We show that generic residual intersections have sufficiently small singular locus. In section 3 we apply this to obtain that the' cycle f~k+l is just the double point cycle of the generic linear projection Pk. Finally in section 4 we interpret vk as the ramification cycle associated to Pk.

We thank the referee for useful suggestions and remarks.

2 R e s i d u a l i n t e r s e c t i o n s

In this section we will prove some facts about generic residual intersections which will be used for the proof of the main result.

(2.1) Let V be an algebraic K-scheme which is Cohen-Macaulay, and W C V a subscheme which is locally a complete intersection of codimension r in V. Consider sections F1 . . . . , F,, E F(V, Jw) with m > r, where J w c Ov is the ideal sheaf of W. Then the residual intersection of V(FI , . . . , Fro) with respect to W is defined as the subscheme B c.C_ V ( F h . . . , Fro) given by the colon ideal

fib := (F l , - . . , F,~): Jw.

The following lemma is immediate from the definition

(2.2) Lemma . With the notation of (2.1) the following hold.

(1) B does not depend on the choice of generators of the ideal (F1, . . . , Fro). (2) B \ W = V(F1 . . . . . fm) \ W.

In the main result of this section we will use the following notation.

[]

b ~ c_ A ~ × V =: Y

(2.3) Let W C V be as in (2.1) and assume that J w is generated by elements gl . . . . , g~ E F(W, flw) (where N is not necessarily equal to r = codimvW). For indeterminates U~, 1 _< #, u < N, we set

N

F.:= U. gv, V=I

and consider for r < m _< N the residual intersection


of V ( F x , . . . , F , , ) with respect to W := A N2 × W. Here A N = AN indicates the N-dimensional affine space. Let q : B m --* V denote the canonical projection.

(2.4) P ropos i t i on . With the above notation the following hold.

(1) (Bm ca W ) \ B 'n+l C_ 13 m is given by the equation Fro+t, i. e

(t3" ca w ) \ B "+~ = ( ~ ca {F~+~ = 0}) \ B m+'.

(2) B m and BmcaW are Cohen-Macaulay, codimv B r" = m, eodimw. B'~NW = 1 and q : B m \ }IV --~ V is a smooth morphism. Moreover V(F1 . . . . . Fro) is the scheme theoretic union of B m and ~) , i.e. (F1, . . . , FN ) is the intersection of the ideal sheaves o r b m and W .

(3) I f V and W are smooth and N > m + 1 then

Sing B ~ C B m+l ¢'IPY

Sing(B m f l w ) g B m+l N W.

Proof. For the proof of (1) observe that by definition B m is the support of the sheaf ffw/(~um=l FI~ Or) and so

Jw I(V \ ~ ' )= ~ F,, o,,\~,-

for every m. Thus B m ca W C B m is given on 1~ \ /3 m+l by the equations FI . . . . . Fm+x = 0. As F1 , . . . , Fm vanish identically on B m, (1) follows.

For the proof of (2) let x e V be a point. If x ¢ W and, say, 9~(x) ~ 0 then B" is given in a neighbourhood of q- l (x) by the equations

N

v.~ + ~ v . . gJg~, 1< ~, < m, V=2

which define a subspace which is smooth over V and has codimension m in P. If x E W then we may assume that 9x,.. . ,9r generate ffw in a neighbourhood of x. For j > r write

r

g J = Y ] aJogo Q=I

so that

o==l j > r

Let D m be the subscheme of • given by the r-minors of the matrix r - 1

i > r l < # < m , l_<L0<r.

Then codimw D~'_ 1 = m - r + 1, and the ideals ffw and (F1 . . . . . Fro) are equal oil Fd \ D~_~, i.e. B '~ f3 YV C_ Dr~ ~.


Thus it follows that (Ft . . . . . Fro) satisfies the Artin-Nagata condition Grn, i.e. l~((Fl,... ,F,,)Ov,~) <_ dimOv, x for all x E V(F1, . . . ,Fro) with dimOv,z < m, see [Hun]. Moreover

c o d i m v B " Q W > _ r e + l ,

codimv B m >_ m >_ r = codimv W.

Hence B m is the geometric m-residual intersection of V(F1, . . . , Fro) with respect to W in the sense of [HU1], see also [FOV], 7.2.1. As 142 is a complete intersection in P it is trivially strongly Cohen Macaulay. By Huneke's criterion [Hun], Theorem 3.1, see [FOV] 7.2.4, we get:

(a ) /3 m is Cohen-Macaulay of codimension m in 12. (b) (F~,..., F,,) = Js~ n Jw. (c) B " Cl W is Cohen-Macaulay of codimension 1 in B TM.

For the proof of (3) we first observe that by Cramer's rule the r-minors of (U~a) are contained in fin.-. Thus D~_ 1 _D/3 "~ fq W. Being a determinantal variety D~m_l is also Cohen-Macaulay of the same dimension as B m Iq W. By (1) these varieties are equal outside B "~+1, which is of codimension m + 1. Hence D~_ 1 = Bran 14;. It follows that the fibers of

(2.4.1) q : B m n ~,V ---* W

are generic determinantal varieties (up to smooth factors). Thus q is a smooth morphism outside the set D~_2 given by the (r - 1)-minors of (U~) ,s ,~ , , .

,_<e<r By (2), Sing B m C_ W. Let x E (~m("lW)\~m+l be a point. ,~oA~ X"r-lr)rn+l ---- ~m+lf-.iW

t the matr ix (U',,~),s,_<,,+, has rank r at x and so (U~o),_~,s,. has rank > r - 1 at x. - r ~ - ,_<Q_<:r l < _ f < r

Thus x is a nonsingular point of B m f'l W proving tile second assertion of (3). As B" Q W is given by one equation at x in B " also B m is smooth at x. U

Taking the generic fibers of the map B "~ ----* A t~ resp. ~) ~ A N2 we obta in the following result.

(2.5) C o r o l l a r y . Let V be an algebraic manifold and W C V a submanifold of codimension r given by equations g l , . . . , glv. For indeterminates U~v, 1 < #, ~, < N, set

L := K(U~),<.,~(_N N

Forr < m < N let Xm C VL := V ®K L be the set of zeros Of F h . . . , F m . Then

X ~ = Wc U B" ,

where B " is a m-codimensionaI Cohen-Macaulay subvariety of VL. Moreover, B 'n \ W is smooth, and there is a Zariski open dense subset U of A " := B " Q WL such that A TM is a smooth Cartier divisor in B m given by the equation Fr~+l at the points of V. ['1

(2.6) Remarks. (1) Slightly more generally assume in (2.5) that V is an algebraic variety and W c V is a subscheme such that V \ W is smooth. Then by the proof


of (2.5) and (2.4) the residual intersection B m of WL with respect to Xm is smooth outside WL.

(2) Also, the Cohen-Macaulayness of B "~ in (2.5) is true under weaker assumptions. I t is sufficient to assume that W is subvariety of an algebraic variety V satisfying the following conditions:

(i) W C_ V is strongly Cohen-Macaulay, (ii) W C V satisfies the Art in-Nagata condition Gin, i.e. for all x E W with

dim Ov,~ < m iz(ffw,=) <_ dim Ov,~,

where # ( . . . ) denotes the minimal number of generators of the ideal in question.

(3) Let the notation and assumptions be as in (2.5). Then the subscheme A m = B m fl WL is just the union of all irreducible components of B m A V(F,~+I) which are contained in Wt.. This follows from the fact that B ~" f3 WL is Cohen-Macaulay by (2.4) (2). Using (2.4.1) we get that the set of singular points of A'* even has codimension 2 ( m - r + 2 ) . Moreover, the singularities e r a m are generic determinantal singularities up to smooth factors.

3 The Stiickrad-Vogel a lgor i thm and double point cycles

(3.1) Let V be an equidimensional algebraic K-scheme. For a line bundle £ over V and d > 0 consider sections

ao . . . . . ad e H° (V , 12).

Let W := {no . . . . . ad = 0} be the set of zeros of no,. • •, cra. In the Stiickrad-Vogel algorithm one constructs from these data cycles v( = vi(a__, V) and fli = Pi(a__, V), i > 0 of dimension i on W, resp. V, as follows.

We introduce indeterminates U(j and set

L := K(Uo)o<i,j<<_d ,

li n : = E j = 0 U~ a t, O < i < d ,

D~ := {ll = 0} .

Then Di is a subscheme of VL := V ®K L of codimension < 1. We start by decom- posing [VL] as a direct sum

[VL] = &~m v + Vdt,. v

where supp(vdim V) C Wt. and no (reduced) component of fldimV is contained in WL. If flk has already been constructed then we write again the intersection fl~ Iq Bali m V - k

as a sum t~k ("1DdimV-k : [~k-1 "-I'- Vk-1

where as above supp(vk-l) C Wz and no component of flk-I is contained in WL. Observe that every component of flk meets Odi mV_k properly by construction, see [~ovl.


(3.2) This construction will be applied to subvarieties X, Y C_ pN in the following way. Let J := J ( X , Y ) C_ p2N+l be the ruled join variety of X and Y, i.e. the variety whose affine cone J in A 2N+2 -~ A lv+l x A N+I is just the product )( x 1/. Let

Ap~ : = { ( z : z ) e p2~+~ : z e pN}

be the diagonal subspace. Then ApN N J ~ X ;'l Y is given by the equations

X o - Y o . . . . . X N - - Y N = 0

in J, where (xo : . . . : XN : Yo : "'" : YN) are the homogeneous coordinates in p2N+l. Applying the above construction to

Y : = J , W : = X N Y , a ~ : = X i - Y i • H ° ( J , Oj(1)),

we obtain cycles fli = ~ i ( X , Y ) , v i = v i (X ,Y) .

The main result of [SVo] is the following generalized Bezout theorem.

(3.3) T h e o r e m . deg X deg Y = ~]i>0 deg v~ + deg fl0. Q

In [SVo] the number deg f~0 was called the multiplicity of the empty set. It admits the following geometric interpretation in terms of the embedded join X Y . Recall that the embedded join variety X Y is by definition the image of J under the rationM map

7r: J - - ~ P~

(x : y) ~--* z - y.

(3.4) Theo rem. (van Gastel [vGa])

deg f~0 = deg X Y . deg (J /XY) ,

where deg J / X Y is the mapping degree of 7r. [:3

In the rest of this section we will give a geometric interpretation of the cycle f~a as the cycle of double points of a generic linear projection. In the following we equip the projective space pN = p~ with homogeneous coordinates X0, . . . , X~.

(3.5) DEFINITION. For indeterminates U~j put

/V

~ , :=~ u, j x j , o<i<~v. j=O

Then ,ko,..., AN are linear forms on P~ where L := K(Uij)o<i,l<N. We call the map

p : = : . . . : - - 9

the r-th generic linear projection.


We remark that for a subscheme Z C pN with dim Z < r the restriction of p to ZL is a regular map. Moreover we note that the )h's and li's from (3.1) are related via

li(X,Y_) = Ai(X) - Ai(Y) for 0 < i < N.

(3.6) DEFINITION. Let X, Y C pN be algebraic varieties of dimension n, m respectively and f : X LI Y - - 1~ be a linear projection. Let

] : S := J(X, Y ) ~ P := J(e ' , P')

be the map induced by f and denote by Ai, C_ P, Axn v C_ 3 the diagonal subspaces. Then the residual intersection of ] -1(A!, ) with respect to Axny is called the double point scheme of f and will be denoted by DI(X, Y).

(3.7) Remark. Usually the double point scheme is introduced in a slightly different way as the residual intersection I ) I (X, Y) of the diagonal X N Y ~ X x Y with respect to ( f x f ) - l ( A ) where A ~ pr ,_, I~ × W is the diagonal. We remark that these two definitions coincide essentially. More precisely, the canonical map

D/CX, y) \ ( i(x)uj(Y)) ---, D:(x, Y)

(x : y) ~ (~, y)

is an isomorphism outside the diagonals where i : X ~ J(X, Y), j : Y ~ J(X, Y ) are the maps x ~-* (x : 0) resp. y ~ (0 : y).

Proof. The reader may easily verify that this map is surjective. For injectivity, consider

(x : y), (x' : y') E DI (X , Y ) \ ( i(X) U j (Y) )

such that (x,y) = (x' ,yl), i.e. x = Ax' and y = ry ' for some A,7- E K*. As ( f (x) : f (y)) , (f(x ') : f (y ')) are in the diagonal it follows that

f(x) = f(y), f(x') = f(y')

(as points in the affine space). Thus by the linearity of f

Af(x') = f(Ax') = f(x) = f(y) = Tf(y'),

forcing A = T. []

(3.8) P r o p o s i t i o n . Let X, Y C pt~ be algebraic varieties of dimension n, m respectively. Let p := Pk : XL U YL ~ p .+ ,n-k- i be the generic linear projection, - 1 < k < d i m X fq Y - 1, and set D := Dp(XL, YL). Then dimD < k + 1 and

[D,(X,..YL)]~+, = ~+,.

Moreover

codimD (Sing D \ AXLW,.L) > cod imj (Sing J \ Ax~nyL).

Flenner-lVlanaresi 281

Proof. The equality of/3k and [D] follows immediately from the fact that

D \ AXLnVL

is given by the vanishing of

I i (X,__Y)=Ai(X)-Ai (E) , O < i < n + m - k - 1 ,

and the construction of/3k+1 The second statement follows from (2.5). El

Applying this proposition to tile case k = - 1 and specializing the indeterminates Uij to generic elements uij in K we obtain the following result.

(3.9) Coro l la ry . Assume that n + m < N. Let p : X U Y ---, E "~+m be a generic linear projection. Then deg tic is the numbers of points in the double point set Dp(X,Y) ofp. t:]

In the following we will say that a variety has an ordinary double point at z if it has analytically two branches which are smooth and meet transversally at z. If in (3.9) X = Y is smooth and z is an ordinary double point of p(X) then z = p(x) = p(y) for some x, y E X. Multiplying z by a constant we may assume that p(z) = p(y) as points of the affine space. Then (x : y), (y : x) e Dp(X,X). Thus the number of points in Dp(X, X) is twice the number of ordinary double points of p(X). Applying (3.9) gives the following corollary.

(3.10) Coro l la ry . Let X C_ ply be a subvariety of dimension n with N > 2n. Let p : X - -~ p2n be a generic linear projection. Then p(X) has at most ordinary double points, and their number, say 6, is given by

28 = deg /3o = deg S e c X . d e g ( J / S e c X )

where J = J(X, X) is the ruled join. [3

(3.11) Example. Assume that C C pN is a nonsingular curve of degree d and genus g. Let p : C -----* p2 be a generic linear projection. Then p(C) has at most ordinary double points whose number & is given by ( d - 1 ) ( d - 2 ) / 2 - g, by Pliicker's formula. Thus

deg /~o

and by Bezout's theorem

= 2 ( ( 4 - ~) (d- 2) 2 e)

- - [ C l ,

deg v0 = d 2 - deg vl - deg/3o = 2d - 2 + 2g.

We remark that this number is just the number of ramification points of a generic linear projection C , pl by Hurwitz's formula.


4 Generic projections and ramification cycle

Let X C pN be a submanifold of dimension n and p : X ---* p2n-k+l be a generic linear projection where 0 _< k < n - 1. In this section we will show that the ramification locus of p has dimension < k and that its associated k-cycle is just the Stiickrad-Vogel cycle Vk. We will do this more generally for two subvarieties X, Y C pN meeting sufficiently nice. First we will precise what this means.

(4.1) DEFINITION. Let X , Y be closed subschemes of the algebraic K-scheme Z. Then the set

Sm(X, Y) := {x e X n Y : X, Y and X N Y are smooth at x}

is called the smooth locus of the pair (X, Y). We will say that X, Y meet smoothly if Sm(X, Y) = X O Y.

For instance, if Y is a submanifold of a manifold X then X and Y meet smoothly. In particular, in the case of a selfintersection, i.e. X = Y, the smooth locus is just the set of smooth points of X.

(4.2) L e m m a . Let X, Y be closed subschemes of an algebraic K-scheme Z. Assume that x E X N Y and X n Y, X and Y are smooth at x. Then

f21xuy,= @ OxnY,=

is a free Oxnv,=-module of rank dim= X + dim= Y + dim {z} - dim= X n Y.

We leave the simple proof to the reader.

D

! (4.3) Remark. Observe that the minimal number of generators of ~xuY,= ® Oxnr,= is always at least

dim= X + dimu Y + dim {x} - dim= X n Y,

by [FVo], (3.5) and (3.7).

(4.4) Example. Assume that X, Y are submanifolds of Z meeting transversally at a point z E X n Y. Then X, Y meet smoothly at x.

For a map of locally free sheaves ~o : £ ~ .T" we let D(~o) be the degeneracy locus of ~o, i.e. the subscheme of X given locally by the maximal minors of ~ in a suitable local basis of E and .T.

Now we are able to introduce the relative ramification locus of a map f : X U Y ~ N where N is a manifold.

(4.5) DEFINITION. Let X, Y C Z be closed subscheme of the algebraic K-scheme Z and f : Z ~ N a morphism into an algebraic manifold with dim N _> dim X + dim Y - dim X n Y. Let

f*(n'~) ® OxnY ~ ~'~o,. ~ oxny.


the induced map and assume that X, Y meet smoothly. Then the subscheme

R(I, x, r ) := D(~)

will be called the relative ramification locus of ~o.

As an example, consider the case that Y is a submanifold of the manifold X. Then R( f , X, Y ) is the set of all y 6 Y such that T~flTv, u : Tv,~ ~ TN, I(u) is not injective. The main result of this section is the following theorem.

(4.6) T h e o r e m . Let X, Y C_ pN be algebraic varieties of dimension n and m, respectively. Let k be an integer such that 0 < k < d i m X N Y - 1 and N >_ n + m - k - 1. Let p : XL O YL ~ p~+m-k-1 be the generic linear projection, see (3.5), and R(p) = R(p, XL, YL) its ramification locus. Then dim R(p) < k, and the associated k-cycle [R(p)]k is just Vk on Sm(XL, YL).

Before proving the theorem we need the following lemmata.

(4.7) L e m m a . Let f : M ~ N be a morphism of algebraic manifolds and A C N a submanifold of codimension m such that

f - I A = X U B ,

where (i) X, B C M are smooth, (ii) A := X A B is smooth, (iii) codimB A = 1, codimM B = m, codimM X = n + 1.

Then there is an exact sequence

f ' ( J a / J ~ ) ® Ox ~-~ Jx/J~ --~ j / j 2 ___, 0

where f f C_ OB is the ideal of A in B. Moreover [ ID(~o)] = [A].

Proof. Let x 6 A be a point and set n := codimMX -- 1. In a neighbourhood of x there are defining equations

g L , . . . , g n , t l , . ' ' , t m - , of B

gi , . . . ,g , , t ' of X.

Then

JxuB = g~ OM + ~_, t % OM. r = l a = l

As JXuB = , .~ " OM and J ~ has locally m generators, the map ~a is given in a suitable local basis, say el . . . . . e,, f , . . . . . fro-, , of f * ( J ~ / ff~) ® O x '~ O ~ by

fi~t't--Ti6Jx/J~, l <i<m-n.


The cokernel of ~o is generated by the residue class of t ' modulo the relations t'ti = O, i = 1 . . . . , m - n. Thus Coker (T) ~ if/,.72. Denoting the n x n unit matrix by E~ the matrix of ~ is given by

0 tm-n ) , , . . .

with respect to the basis e b . . . , e. , f l . . . . , f ro- . of f ' ( f f a / f f ~ ) ® O x resp..ql . . . . . .~., t~ of f f x / f f ~ . Hence D(~o) is the subspace of X given by the equations tl . . . . . tn-m = 0, and so it is just A. []

We recall the following result from [FVo], (3.7).

(4.8) L e m m a . Let X , Y C_ Z be closed subschemes of the algebraic K-scheme Z. Let f fxnY ~ Ox×y be the ideal of X n Y which we consider as a subscheme of X x Y via the diagonal embedding. Then

3x~Y / 5 ~ y ~- fllxov ®oxo~ Ox~v.

Proof of theorem (4.6). The generic linear projection p is given by

D

p = (,~o : . . . : , ~ , + m - ~ - 1 )

where Xi := ~ = 0 Uij xj is as in (3.5). Let P be the join j ( ~ + m - k - l , i m + m - k - t ) which is a projective space of dimension 2n + 2m - 2k - 1. There is an induced map of joins

: J := J ( X , Y) - -~ F.

Let AxnY "~ X VI Y ~ J, A '~ l ~+m-k-1 ~ p be the diagonals and 3xnY ~ O j , ~:7a C Oe their ideal sheaves.

Let Bk+l be the residual intersection of iS-l(Ac) with respect to Ax~nY L. Using (2.5) we get the following:

(a) Bk+l is Cohen-Macaulay on Sm(Xc, YL) and even smooth on Reg(JL) \ A x~.nYL.

(b) As : - Bj,+I VI ~x~nYL is generically smooth. (e) The codimension of Ak in Bk+l is 1 in the points of Sm(X,Y), and

codims Bk+l = n + m - k.

Using the fact that iS-~(AL) is given by the equations

N

l, = ~, ® 1 - 1 ® Ai = ~--~ U o ( X j ® I - I ® X j ) , $=0

we get from the Stiickrad-Vogel algorithm that

O < i < n + m - k - 1 ,

& + , = [ B , + , ] , + ,

vk = {A, ]~ .


Let )( , Y c A N+I be the affine cones of X, Y and i-I l . the differential module - - X u Y / K

of their (scheme theoretic) union. Using (4.8) it follows that

2 ~u~ ' / t : ® Oxnv = `Txnv / J~nv ,

where ~ denotes the associated sheaf on the projective space. Similarly, as A "+m-k = (p.+,~-k-~)- on p.+, . -~- i

Consider the following diagram of vector bundles on Sm(XL, YL) where we write Z := X N Y for short:

0 -~ p'a~,,+.,_~_,/,.~ -~ p'-laA;+~_~/~ -~ p ' ( , T A J j L ) - , o ~ --, o

dp 1 d~ = ~o ~ "~ 1

0 ~ ~x~.uv~/z. ® OzL "-, £z,~u~.,./t. ® OzL = J z , / J ~ , . ---, Oz~. ---, 0 .

Here the rows are just the Euler sequences. It follows that

D(dp) -- D(di~),

and so the ramification locus Rip, X , Y) is given by the maximal minors of the induced map ~ : ~, ~ ALl ~ ) ~ `7ZL/J~L (with respect to some local basis). Outside of Bk+l the map ~o is an isomorphism, and so R(p, X, Y) is contained in Bk+l Q Az L, By (4.7) on an open subset of Sm(XL, YL) there is an exact sequence

p (:TaL/J~,~) ® OzL ~ J z L / ZL ---'* j i j 2 ~ 0

where ,7 C_ OBk÷~ is the ideal sheaf of Ak in Bk+l. Moreover, [ D(~)] -- [Ak]. El

Specializing to the case of selfintersection, i.e. X = Y, we have the following result.

(4.9) Coro l l a ry . Let X C pN be an algebraic variety of dimension n. Let k be an integer such that O < k < n and N >_ 2 n - k - 1 . Let P : XL ~ p~n-k-l be the generic linear projection and Rip) its ramification locus. Then dim Rip ) <_ k, and the associated k-cycle [R(p)]k is just vk on Reg(X).

The following two results were suggested to us by the referee.

(4.10) Coro l l a ry . Let X C_ pg be an algebraic variety of dimension n and Y C_ Reg X a submanifold of dimension m. Then vk(Y, X) is just the intersection of vk(X, X ) with Y .

We will give a geometric application along the line of thought of Johnson [Joh].

(4.11) Coro l l a ry . Suppose that X and Y are smooth subvarieties o fP N such that also X Q Y is nonsingular. Let p : X U Y ---* pn+m-k-I be a generic projection. I f the tangent map

T~p : Txuv,= "* Tp,+..-~-l,p(=)

has maximal rank for all x E X Q Y then p has no double points on X U Y , i.e. i f p(x) = p(y) ]or some (x ,y) E X x Y then x = y.

Proof. By (4.6) we know that the cycle Vk vanishes. We need to show that then also fl~+l = O. But this is a consequence of [FGV]. D


R e f e r e n c e s

[AMa]

[FOV]

Achilles, R.; Manaresi, M.: An algebraic characterization of distinguished varieties of intersection. Rev. Roumaine Math. Pures Appl. 38, 569-578 (1993).

Flenner, H.; O'Carroll, L.; Vogel, W.: Joins and intersections. In prepara- tion.

[FCV]

[FVo]

[Full

[veal

[H~nl

[HUll

[Hun]

[Joh]

[SVo}

Flenner, H.; Gastel, L.J. van; Vogel, W.: Joins and intersections. Math. Ann. 291,691-704 (1991)

Flenner, H.; Vogel, W.: Limits of joins and intersections. In: "Higher dimensional complex varieties". Proceedings of the International Conference, Trento (1994), pp. 209-220. Walter de Gruyter, Berlin-New York 1996.

Fulton, W.: Intersection theory. Erg. Math. Grenzgeb., 3. Folge, Bd. 2, Springer-Verlag, Berlin Heidelberg New York, 1984.

Gastel, L.J. van: Excess intersections and a correspondence principle. In- vent. Math. 103, 197-221 (1991).

Hansen, J.: Double points of compositions. J. Reine Angew. Math. 352, 71-88 (1984).

Huneke, C.; Ulrich, B.: Residual intersections. J. Reine angew. Math. 390, 1-20 (1988).

Huneke, C.: Strongly Cohen-Macaulay schemes and residual intersections. Transactions of the AMS 277, 739-763 (1983)

Johnson, K.W.: Immersion and embedding of projective varieties. Acta Math. 140, 49-74 (1978).

Stiickrad, J.; Vogel, W.: An algebraic approach to the intersection theory. In: The curves seminar at Queen's Vol. II, 1-32. Queen's papers in pure and applied mathematics, No. 61, Kingston, Ontario, Canada, 1982.

H. Flenner Fakult~it fiir Mathematik R.uhr-Universit~t, Geb. NA 2]72 44780 Bochum Federal Republic of Germany e-mail: Hubert.Flenner~rz.ruhr-uni-bochum.de

M. Manaresi Dipartimento di Matematica Universit£ 40127 Bologna Italy e-mail: [email protected]

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Intersections of projective varieties and generic projections