INTERSECTION THEORY CLASS 13

Transcription

1 INTERSECTION THEORY CLASS 13 RAVI VAKIL CONTENTS 1. Where we are: Segre classes of vector bundles, and Segre classes of cones 1 2. The normal cone, and the Segre class of a subvarety 3 3. Segre classes behave well wth respect to proper and flat morphsms 3 1. WHERE WE ARE: SEGRE CLASSES OF VECTOR BUNDLES, AND SEGRE CLASSES OF CONES We frst defned Segre class of vector bundles over an arbtrary scheme X. If E s a vector bundle, we get an operator on class on X. We defne t by projectvzng E, so we have a flat and proper morphsm PE X, pullng back α to PE, cappng wth O(1) a certan number of tmes, and pushng forward. Hence we get s (E) : A k X A k X, and for example we checked the non-mmedate fact that s 0 (E) s the dentty. (Recall s 0 nvolved pullng back, cappng wth precsely ranke 1 copes of O(1), and then pushng forward.) Note that s k (E) = s k (E 1), as the Whtney product formula gves s(e 1) = s(e)s(1) = s(e). We want to generalze ths to cones. Here agan s the defnton of a cone on a scheme X. Let S = 0 S be a sheaf of graded O X -algebras. Assume O X S 0 s surjectve, S 1 s coherent, and S s generated (as an algebra) by S 1. Then you can defne Proj(S ), whch has a lne bundle O(1). Proj(S ) X s a projectve (hence proper) morphsm, but t sn t necessarly flat! (Draw a pcture, where the cone has components of dfferent dmenson.) Flat morphsms have equdmensonal fbers, and cones needn t have ths. A couple of mportant ponts, brought out by Joe and Soren. I ve been mprecse wth termnology. Although one often sees phrases such as the cone s C = Spec(S ), we lose a lttle nformaton ths way; the cone should be defned to be the graded sheaf S. The sheaf can be recovered from C X Y along wth the acton of the multplcatve group O X ; the nth graded pece s the part of the algebra where the multplcatve group acts wth weght n. Example 1: say let E be a vector bundle, and S = Sym (E ). Then Proj S = PE. Example 2: Say T = Sym (E 1) = S S 1 z, so (better) T = S [z]. Then Proj T = PE. Example 3: Date: Wednesday, November 3,

2 Proj(S [z]) = C Proj(S ) = Spec S Proj(S ). The argument s just the same. The rght term s a Carter dvsor n class O Proj(S [z])(1). Example 4: The blow-up can be descrbed n ths way, and t wll be good to know ths. Suppose X s a subscheme of Y, cut out by deal sheaf I. (In our stuaton where all schemes are fnte type, I s a coherent sheaf.) Then let S = I, where I s the th power of the deal I. (I 0 s defned to be O X.) Then Bl X Y = Proj S. A short calculaton shows that the exceptonal dvsor class s O( 1). The exceptonal dvsor turns out to be Proj I n /I n+1. (Note that ths s ndeed a graded sheaf of algebras.) As I n I n /I n+1 s a surjectve map of rngs, ths ndeed descrbes a closed subscheme of the blow-up. (Remember ths formula t wll come up agan soon!) So the same constructon of Segre classes of vector bundles doesn t work: there s no flat pullback to Proj(S ). So what do we do? Idea (slghtly wrong): We can t pull classes back to Proj(S ). But there s a natural class up there already: the fundamental class. So we defne s(c)? = q ( 0 c 1 (O(1)) [ProjC]) where q s the morphsm Proj C X. Instead, as Segre class of vector bundles are stable wth respect to addng trval bundles, we defne s(c) := q ( 0 c 1 (O(1)) [Proj(C 1)]) where q s the morphsm Proj(C 1) X. Why s addng n ths trval factor the rght thng to do? Partal reason: f C s the 0 cone,.e. S = 0 for > 0, then Proj C s empty, but Proj C 1 s not; we get dfferent answers. But f you add more 1 s, you wll then get the same answer: s(c 1 1) = s(c). (Exercse: show that s(c 1) = s(c).) Note: s has peces n varous dmensons. Last tme I proved: Proposton. (a) If E s a vector bundle on X, then s(e) = c(e) 1 [X], where c(e) s the total Chern class of X, r = rank(e). c(e) = 1 + c 1 (E) + + c r (E). (I would wrte s(e) = s(e) [X], but the two uses of s(e) are confusng!) Ths s bascally our defnton of Segre/Chern classes. (b) Let C 1,..., C t be the rreducble components of C, m the geometrc multplctes of C n C. Then s(c) = t =1 m s(c ). (Note that the C are cones as well, so s(c ) makes sense.) In other words, we can compute the Segre class pece by pece. 2

3 2. THE NORMAL CONE, AND THE SEGRE CLASS OF A SUBVARIETY Let X be a closed subscheme of a scheme Y (not necessarly lc = local complete ntersecton), cut out by deal sheaf I. I/I 2 s the conormal sheaf to X; t s a sheaf on X. (Why s t a sheaf on X? Locally, say Y = Spec R, and X = Spec R/I. Then ths s the R-module I/I 2. The fact that I sad that t s an R-module makes t a pror a sheaf on Y. But note that t s also an R/I module; the acton of I on I/I 2 s the zero acton.) If X s a local complete ntersecton (regular mbeddng), then ths turns out to be a vector bundle. Consder n=0 In /I n+1. (Recall that Proj of ths sheaf gves us the exceptonal dvsor of the blow-up.) Defne the normal cone C = C X Y by C = Spec I n /I n+1. Defne the Segre class of X n Y as the Segre class of the normal cone: n=0 s(x, Y) = s(c X Y) A X. If X s regularly mbedded (=lc) n Y, then the defnton of s(x, Y) s s(x, Y) = s(n) [X] = c(n) 1 [X]. The followng geometrc pcture wll come up n the central constructon n ntersecton (the deformaton to the normal cone). X A 1 Y A 1. Then blow up X 0 n Y A 1. The deal sheaf of X 0 s I[t], where t s the coordnate on A 1. Thus the normal cone to X 0 n Y A 1 s C X Y[t]. Hence the exceptonal dvsor s Proj(C X Y[t]) (draw a pcture). Insde t s the Carter dvsor t = 0, whch s Proj(C X Y). 3. SEGRE CLASSES BEHAVE WELL WITH RESPECT TO PROPER AND FLAT MORPHISMS Ths s the key result of the chapter. Proposton. Let f : Y Y be a morphsm of pure-dmensonal schemes, X Y a closed subscheme, X = f 1 (X) the nverse mage scheme, g : X X the nduced morphsm. (a) If f proper, Y rreducble, and f maps each rreducble component of Y onto Y then g (s(x, Y )) = deg(y /Y)s(X, Y). (b) If f flat, then g (s(x, Y )) = s(x, Y). Let me repeat why I fnd ths a remarkable result. X s a pror some nasty scheme; even f t s nce, ts codmenson n Y sn t necessarly the same as the codmenson of X n Y. The argument s qute short, and shows that what we ve proved already s qute sophstcated. 3

4 As a specal case, ths result shows that Segre classes have a fundamental bratonal nvarance: f f : Y Y s a bratonal proper morphsm, and X = f 1 X, then s(x, Y ) pushes forward to s(x, Y). Proof. Let me assume that Y s rreducble. (It s true n general, and I may deal wth the general case later.) Let me frst wrte the dagram on the board, and then explan t. O Proj(C 1)(1) = G O Proj(C 1) (1) O Proj(C 1) (1) q X q g X Proj(C 1) G Proj(C 1) Carter dv. Carter dv. Bl X 0(Y A 1 ) F Bl X 0 (Y A 1 ) We blow up Y A 1 along X 0, and smlarly for Y and X. The exceptonal dvsor of Bl X 0 (Y A 1 ) s Proj(C 1), and smlarly for Y and X. The unversal property of blowng up Y A 1 shows that there exsts a unque morphsm G from the top exceptonal dvsor to the bottom. Moreover, by constructon, the exceptonal dvsor upstars s the pullback of the exceptonal dvsor downstars (that s the statement about the two O(1) s n the dagram). Let q be the morphsm from the exceptonal dvsor Proj(C 1) to X, and smlarly for q. That square commutes: q G = g q (bascally because that morphsm G was defned by the unversal property of blowng up). Now f [Y A 1 ] = d[y A 1 ] (where I am slopply usng the name f for the morphsm Y A 1 Y A 1 ). Ths s computed on a dense open set, so blow-up doesn t change ths fact: F [Bl X 0Y A 1 ] = d[bl X 0 Y A 1 ]. Now we ve shown that proper pushforward commutes wth ntersectng wth a (pseudo-)carter dvsor. Hence G [Proj(C 1)] = d[proj(c 1)]. 4

5 Now I m gong to prove (a), and I m gong to ask you to prove (b) wth me, so pay attenton! g s(x, Y ) = g q c 1 (G (O(1)) [P(C 1)]) (by def n) = q G ( ) c 1 (G (O(1)) [P(C 1)]) (prop. push. commute) ) ( = q c 1 ((O(1)) d[p(c 1)]) (proj. form. ) (.e. c 1 commutes wth prop. pushforward) = ds(x, Y) (by def n) Now (b) s smlar: g s(x, Y) = g q c 1 ((O(1)) [P(C 1)]) (by def n) = q G c 1 ((O(1)) [P(C 1)]) (push/pull commute) = q c 1 ((G O(1)) G [P(C 1)]) = s(x, Y) (by def n) We mmedately have: Corollary. Wth the same assumptons as the proposton, f X s regular mbedded (=lc) n Y, wth normal bundle N, then g (c(n ) 1 [X ]) = deg(y /Y)s(X, Y). If X Y s also regularly mbedded, wth normal bundle N, then g (c(n ) 1 [X ]) = deg(y /Y)(c(N) 1 [X]). To see why the frst part mght matter: Suppose X Y s a very nasty closed mmerson. Then blow up Y along X, to get Y wth exceptonal dvsor X. Then X s regularly mbedded (lc) n Y t s a Carter dvsor! Ths s the content of the next corollary. 5

6 Corollary. Let X be a open closed subscheme of a varety Y. Let Ỹ be the blow-up of Y along X, X = PC the exceptonal dvsor, η : X X the projecton. Then s(x, Y) = k 1( 1) k 1 η ( X k ) = 0 η (c 1 (O(1)) [PC]) In that frst equaton, the term X k should be nterpreted as the kth self ntersecton of the Carter dvsor X, also known as the exceptonal dvsor. E-mal address: vakl@math.stanford.edu 6