INTERSECTION THEORY CLASS 17 RAVI VAKIL CONTENTS 1. Were we are 1 1.1. Reined Gysin omomorpisms i! 2 1.2. Excess intersection ormula 4 2. Local complete intersection morpisms 6 Were we re oin, by popular demand: Grotendieck Riemann-Roc (15); comparison to Borel-Moore omoloy (capter 19). 1. WHERE WE ARE We deined te Gysin pullback i! and a rater eneral intersection product. Let i : Y be a local complete intersection o codimension d. Y is arbitrarily orrible. Suppose V is a sceme o pure dimension k, wit a map : V Y. Here I am not assumin V is a closed subsceme o Y. Ten deine W to be te closed subsceme o V iven by pullin back te equations o in Y: W cl. imm. V (notice deinition o ). cl. imm. Y Te cone o in Y is in act a vector bundle (as Y is a local complete intersection); call it N Y. Te cone C W Y to W in Y may be quite nasty; but we saw tat C W Y N Y. Ten we deine V = s [C W V] were s : W N Y is te zero-section. (Recall tat te Gysin pullback lets us map classes in a vector bundle to classes in te base, droppin te dimension by te rank. Alebraic black box rom appendix: as V is purely k-dimensional sceme, C W V is also.) Last time I proved: Date: Wednesday, November 17, 2004. 1
Proposition. I ξ is te universal quotient bundle o rank d on P( N /Y 1), and q : P( N /Y 1) W is te projection, ten V = q (c d (ξ) [P(C W/V 1)]). and Proposition. V = {c( N /Y ) s(w, V)} k d. (Here { } k d means take te dimension k d piece o.) and stated (witout proo): Proposition. I d = 1 ( is a Cartier divisor on Y), V is a variety, and is a closed immersion, ten V is te intersection class we deined earlier ( cuttin wit a pseudodivisor ). 1.1. Reined Gysin omomorpisms i!. Let i : Y be a local complete intersection o codimension d as beore, and let : Y Y be any morpism. Y As beore, C = C Y N Y. Deine te reined Gysin omomorpism i! as te composition: A k Y σ A k C A k N s A k d. Note wat we can now do: we used to be able to intersect wit a local complete intersection o codimension d. Now we can intersect in a more eneral settin. We ll next sow tat tese omomorpisms beave well wit respect to everytin we ve done beore. Tese are all important, but similar to wat we ve done beore, so I ll state te various results. I ll just sporadically ive proos. Handy act: Say we want to prove sometin about i! [V]. Consider Y V V Ten Y Y i! [V] = c( N /Y ) s( V, V). Reason we like tis: we already know Cern and Sere classes beave well. So we can reduce calculations about i! to tins we ve already proved. Reason or act: Calculate 2
V usin V V Y We et c( N) s( V, V). Pus tis orward to : (c( N) s( V, V)) = c( N) s( V, V)) usin te projection ormula. We now ave to sow tat tis really ives i! [V]. (Fulton uses tis second version as te oriinal deinition.) Omitted. Reined Gysin commutes wit proper pusorward and proper pullback. Consider te iber diaram i Y q p i Y i Y were i is a locally closed intersection o codimension d. (a) I p is proper and α A k Y, ten i! p (α) = q (i! α) in A k d. (Caution: i! means two dierent tins ere!) (b) I p is lat o relative dimension n, and α A k Y, ten i! p (α) = q (i! α) in A k+n d. Proo. (a) We may assume α = [V ] (on Y ). Let V = p(v ) (on Y ). i! p α = de(v /V){c( N /Y ) s( V, V)} k d previous proposition = {c( N /Y ) q s( V, V )} k d Sere classes pus orward well = q {c(q N /Y ) s( V, V )} k d projection ormula = q i! [V ] Compatibility. I i is also a local complete intersection o codimension d, and α A k Y, ten i! α = (i )! α in A k d. It suices to veriy tat N Y = N Y. Reason: I I and I are te respective ideal seaves, te canonical epimorpism (I/I 2 ) I /(I ) 2 must be an isomorpism. (Details omitted. is locally cut out in Y by d equations. is cut out in Y by (te pullbacks o) te same d equations.) 3
1.2. Excess intersection ormula. Consider te same iber diaram as beore i Y q p i Y i Y were now i is still a locally closed intersection o codimension d, and i is also a locally closed intersection, o possibly dierent dimension d. Let N and N be te two normal bundles; as beore we ave a canonical closed immersion N N. Let E = N/N be te quotient vector bundle, o rank d = d d. For any α A k Y, note tat i! (α) and (i )! (α) dier in dimension by e. Wat is teir relationsip? Answer: Excess intersection ormula. For any α A k Y, i! (α) = c e (q E) (i )! (α) in A k d. (Proo sort but omitted.) Immediate corollary. Specialize to te case were te top row is te same as te middle row, and i is an isomorpism: i Y i Y Ten i! α = c d ( N) α. Specialize aain to = Y = to et te sel-intersection ormula: i i α = c d (N) α. Intersection products commute wit Cern classes. intersection o codimension d, i Y Let i : Y be a locally closed i Y a iber square, and F a vector bundle on Y. Ten or all α A k Y and all m 0, in A k d m ( ) Proo omitted. i! (c m (F) α) = c m (i F) i! α Reined Gysin omomorpisms commute wit eac oter. Let i : Y be a locally closed intersection o codimension d, j : S T a locally closed intersection o codimension e. Let Y be a sceme, : Y Y, : Y T two morpisms. Form te iber 4
diaram Y S. j Y i Ten or all α A k Y, j! i! α = i! j! α in A k d e. i Y T Proo (lon!) omitted. Idea: by blowin up to reduce to te case o divisors, as we did wen we sowed tat te intersection o two divisors was independent o te order o intersection, lon ao. Functoriality. Te reined Gysin omomorpisms or a composite o locally closed intersections is te composite o te reined Gysin omomorpisms o te actors. Consider a iber diaram i Y j Z i Y I i (resp. j) is a locally closed intersection o codimension d (resp. e), ten ji is a locally closed intersection o codimension d + e, and or all α A k Z, (ji)! α = i! j! α in A k d e. Proo omitted. Similarly: Second unctoriality proposition. Consider a iber diaram j Z. i Y p Z i Y p Z. (a) Assume tat i is a locally closed intersection o codimension d, and tat p and pi are lat o relative dimensions n and n d. Ten i is a locally closed intersection o codimension d, p and p i are lat, and or α A k Z, (p i ) α = i! p α in A k+n d. (b) Assume tat i is a locally closed intersection o codimension d, p is smoot o relative dimension n, and pi is locally closed intersection o codimension d n. Ten or all α A k Z, (pi)! α = i! (p α) in A k+n d. 5
Sort proo, omitted. 2. LOCAL COMPLETE INTERSECTION MORPHISMS A morpism : Y is called a lci morpism o codimension d i it actors into a locally closed intersection P ollowed by a smoot morpism p : Y. Examples: amilies o nodal curves over an arbitrary base; amilies o suraces wit mild sinularities. Reason we care: oten we want to consider amilies o tins deeneratin. We won t need tis in te next two weeks, but it s wort at least ivin te deinition. For any lci morpism : Y o codimension d, and any morpism : Y Y, we ave te iber square Y Y We want to deine a reined Gysin omomorpism! : A k Y A k d. Here s ow. Factor into p i were p : P Y is a smoot morpism o relative dimension d + e and i : P is a local complete intersection o codimension e. Ten orm te iber diaram i P p Y i P p Y. Ten p is smoot (smoot morpisms beave well under base cane), and we deine! α = i! ((p ) α) (smoot morpisms are lat; tis is part o te deinition). Proposition (a) Te deinition o! is independent o te actorization o. (!!!) (b) I is bot lci and lat, ten! =. (c) Te assertions earlier (pusorward and pullback compatibility; commutativity; unctoriality) or locally closed intersections are valid or arbitrary lci morpisms. Tere is also an excess intersection ormula, tat I won t boter tellin you precisely. Because (a) seems surprisin, and te roo is sort, I ll ive it to you. I i 1 P 1 p 1 Y is anoter actorization o, compare tem bot wit te diaonal: P 1 p 1 (i,i 1) P Y P 1 Y. p P 6
Use te second unctoriality proposition (b). Ten (b) ollows rom (a). (c) is omitted. E-mail address: vakil@mat.stanord.edu 7