Algebraic Geometry (Intersection Theory) Seminar

Algebraic Geometry (Intersection Theory) Seminar Lecture 1 (January 20, 2009) We ill discuss cycles, rational equivalence, and Theorem 1.4 in Fulton, as ell as push-forard of rational equivalence. Notationise, e ill let scheme mean an algebraic scheme over a field, variety ill mean a reduced and irreducible scheme, and a point ill mean a closed point. Let \ be a scheme and Z \ a subvariety. Then b@ßb œ b8ßb here 8 is a generic point of Z. For a variety \, let V \ be the field of rational functions. Let \ be a variety ith Z \ a subvariety of codimension one. Then bzß\ has dimension, and V bzß\ œ V \. For any! Á < V \ œ V bzß\ ith < œ +Î, for +ß, bzß\. For any! Á = b, j b ÎW. Zß\ Zß\ Definition. Let \ be a variety and Z \ a subvariety ith codimension. Then there is a ell-defined homomorphism ord Z À V \ Ä such that ( for < V \ œ V b, < œ +Î, ith +ß, b ): Zß\ (i) ord < œ ord + ord,, (ii) ord Z Z Z Z + œ j b Î+, and Zß\ (iii) ord, œ j b Î,. Z Zß\ Example. If \ is a veriety hich is regular in codimension one, this means for any Z \ a subvariety ith codimension, b Zß\ is a discrete valuation ring. Cycles and rational equivalence Definition. Let \ be a scheme. A 5 -cycle on \ is a finite formal sum 8 cz d ß 8 here Z \ are 5 -dimensional subvarieties of \. Let ^ \ œ e 8 cz d lz are a 5-dimensional subvarieties f, so 5 Any element α œ eαf 5! ^ \ œ 9 ^ 5 \. 5! ^ O 5 is called a cycle. Zß\

For \ a scheme, let \ ßÞÞÞß\ > be irreducible components of \. For each \, e have geometric multiplicity 7 œ j b\ ß\. Notice \ has a generic points ( ( b b is an Artinian ring). \ ß\ œ ( > Notation. We define c\ d œ 7 c\ d ^ \ (this is a cycle). œ We ant to give an equivalence relation on cycles. Let \ be a scheme and [ \ a 5 -dimensional subvariety. For < V [, e define cdiv < d œ ord < czd. cod Z [ [ Z œ Notice a 5 -cycle α is rationally equivalent to!, ritten α µ!, if there are a finite number of 5 -dimensional subvarieties [ \ and < V [ such that α œ cdiv < d. All the cycles equivalent to! are ritten Rat 5 \. We can no define the cycle class, E 5\ œ ^ 5\Î Rat 5\, ith E \ œ 9 E \ œ 9 ^ \Î Rat \. 5 5 5 5! 5! Examples. (1) E 5 \ z E 5 \ red, since \ and \ reduced have the same subvarieties. (2) If \ œ \ \ # ÞÞÞ \ >, then ^ \ œ 9 > ^ \ and E \ œ 9 > E \. œ œ () If dim \ œ 8, then E 8\ œ ^ 8\, then there are no 5 -dimensional subvarieties, so there is nothing to mod out by. (4) If \ and \# are subschemes of \, then is an exact sequence. E 5 \ \ # Ä E 5 \ ŠE 5 \# Ä E 5 \ \ # Ä! Z

(5) If Z is an irreducible component of \, then for any cycle class α E \, e define the coefficient of Z in α to be the coefficient of czd in any cycle hich represents α (since α œ 8 cz d). (6) If \ œ spec O, then E \ œ ^ \ œ (just one point).!! c B d 5 cb d! c: d (7) If \ œ, then E \ œ ^ \ œ and E \ œ Pic œ. (8) Let Z be a 5 -dimensional variety. Assume e have 0 À Z Ä a dominant function ( 0 Z œ, or other ay to think about it is that 0 maps a generic point of Z to a generic point of, or even 0 V Z ). Let's look at the fibers 0! and 0 (i.e., if œ B! À B, e have! œ À! and œ! À ). Then 0! and 0 are purely 5- dimensional subschemes, so c0! 0 d œ cdiv 0 d. Push-forard of cycles Let 0 À \ Ä ] be a proper morphism of to schemes. Then for a unique α ^ \, e ant to understand 0 α ^ ]. Assume α is a 5-cycle ith α œ 8cZd ith Z a 5-dimensional subvariety. We ant 0 czd œ? ^ ] to have some (covariant) functorial property relating \ and ]. If 0 À \ Ä ] and 0 À Z Ä [ œ 0 Z ( 0 Z is a subvariety of ] ). We ant (1) dim [ dim Z means 0cZd œ!. (2) dim [ œ dim Z means V Z ä V [ has degree cv [ À V Z d We then call 0 czd œ cv Z À V [ d c[ d (so 0 α is ell-defined). Theorem. [1.4] If 0 À \ Ä ] is a proper morphism and α is a 5-cycle on \ hich is rationally equivalent to!, then 0α is rationally equivalent to zero on ]. Proof. We rite α œ cdiv < d for < V Z for some 5 -dimensional subvariety of \. That is, α œ cdiv < d œ 8cZd. codim Z Z Z œ Z

For this cycle, e can assume \ œ Z and ] œ 0 Z. We can assume \ is a variety and ] is a variety and 0 À \ Ä ] is a proper surjection. No e use the belo proposition. Proposition. [1.4] Let let < V \. Then 0 À \ Ä ] be a proper, surjective morphism and (a) 0 cdiv < d œ! if dim ] dim \. (b) 0 cdiv < d œ cdiv R < d if dim ] œ dim \, < here R < is determinant of V \ Ä V \ as an V ] -linear morphism ( R V V ] ). Proof. Case 1. Let ] œ spec O for O a field. Let \ œ 5 ith 0 À 5 Ä spec O. Since V \ œ O >, and < V \ œ O > ( < œ +Î, ith +ß, O c> d), e can assume < O c> d is an irreducible polynomial (since cdiv < d œ cdiv + d cdiv, d, so if e prove it for cdiv + d and cdiv, d (and e kno + and, are polynomials), then e have shon it for cdiv < d). Furthermore, let T œ < be a maximal prime ideal ( T \ œ 5 is a closed point). In this case, ord T < œ (since b: œ O c> d < ), and ord U < œ! for U O c> d ith U Á T. Finally, consider the point T 5. Then T œ ˆ > O > (it corresponds to the prime ideal generated by > in the ring O > ). The ay to visualize this is that O c> d is for everything but, and O > is for everything but!. Call degree < œ.. Then < œ ˆ. 2ˆ > >, some poly. For # # example, < œ > œ ˆ ˆ. Then on, e have > > # cdiv < d œ c: d. ct d and 0 cdiv < d œ cv T À Od. cv T À Od œ.. œ!, ith V T œ O c> dî<. Case 2. If 0 À \ Ä ] is finite, set ith O œ V ] and P œ V \ and O ä P. Also, assume all varieties in cdiv < d map to a variety [ ] of codimension. Let E œ b [ß] (the generic point of this image) and 7 œ 7 [ß]. Consider the commutative diagram 5

Spec P Ä \ ] Spec E Ä \ Æ Æ Æ 0 Spec O Ä Spec E Ò ] ith E ä F ä P and dimf œ dim E œ. Finally, F has finitely many maximal ideals 7 such that 7 E œ 7. There is a one-to-one correspondence beteen, ez \l0 Z œ [ f and { 7 Fl7 E œ 7} Þ Also, from the diagram, F Œ O œ P and F Œ O œ P. No assume < F. Then computationally, div < œ ord < cz d, and Z E Z 0 cdiv < d œ ord < cv Z À V [ dc[ d Z Z 7 œ j F Î< cv Z À V [ dc[ d. Z 7 No e use to lemmas from the back of the book (A.2. and A.2.2), that state if 7ßE Ä F ß7 is a push-forard, 7 je F7 Î< œ cf7 Î7 À EÎ7 d j F7 Î< œ ord R < c[ d œ = ord R < c[ d. Z Alternate Definition of Rational Equivalence E Let Z ä \ Ä ith Z a 5 -dimensional variety and \ Ä \. Then the fibers 0! \ e! f and 0 \ e f are purely 5-dimensional subvarieties of Z. Then if 0! \ e! f, e can define div 0 œ c0! d c0 d and : div 0 œ : c0! d : c0 d here : c0! d œ Z! and : c0 d œ Z. Proposition 1.6. A cycle α in ^ 5 \ is rationally equivalent to zero if and only if there are 5 -dimensional subvarieties Z ßÞÞÞßZ > of \ such that projections from Z to are dominant ith > α œ cz! d cz d œ E. E

Proof. We have α œ cdiv < d ( < V [, [ a 5 -dimensional sub- < variety of \). Then consider [ Ä and e have Z ä \ Ä and divc< d œ : cdiv 0 d œ cz! d cz d (here 0 À Z Ä ). Lecture 2 (January 27, 2009) Recall from last time that for \ a scheme, 8 cz d is a 5-cycle, and ^ \ 5 is ^ \ œ e 8 cz d lz are 5-dimensional subvarieties f. 5 Furthermore, ^ \ œ 9 ^ 5\. For [ X 5 -dimensional, < V [, 5! cdiv < d œ ord czd. cod [ Zœ Z < Furthermore, recall that RatV\ œ eeldiv < f ( < µ!). We defined ith E 5\ œ ^ 5\Î Rat5\ E \ œ 9 E 5\. 5! Theorem 1.4. For 0 À \ Ä ] a proper morphism, e define 0 À ^ \ Ä ^ ] \ 5 5 by 0 czd œ deg ZÎ[ c[ d ([ œ 0 Z ). Then for α µ! on \, 0 α µ! on ]. We ill motivate this theorem by seeing its application to Bezout's Theorem on plane curves. Bezout's Theorem on plane curves For JßK plane curves in # ith deg J œ 7ß deg K œ 8, if JßK intersect ith no common roots, then TßJ K œ 7 8, the intersection multiplicity of J and K at T. Assume J is irreducible. Then for all K plane curves, V J, so TßJ K TßJ K œ ord ˆ. K K T K K

Then for all J, V K, Hence, J J TßJ K TßJ K œ ord ˆ. TßJ K œ TßJ K Given J a plane curve, < V J, We ant to sho ord T < œ!. cdiv < d œ ord < ctd. Take 1 À J Ä. Then 1 cdiv < d œ ord <. c1td in, : T T J J (ith. the degree of J ) and notice ord < œ cdiv ; d (; V ).\ Alternate Definition of Rational Equivalence Proposition 1.6. If + µ! in ^ 5 \ if and only if α œ cz! d cz d : Z \ ith 0 À Z Ä dominant ith T À \ Ä \, satisfying Z! œ TJ! and Z œ TJ. Theorem 1.7. If 0 À \ Ä ] is a flat morphism (of relative dimension 8), define 0 À ^ 5] Ä ^ 5 8\ by 0 czd œ c0 Zd ( Z a variety). Then for α µ! in ^ 5], 0 α µ! in ^ \ 5 8. Note. Ramin says: Keep in mind, for flat morphisms, it essentially means that the dimensions of fibers are constant. This is hat it means to be flat, it makes these morphisms nice in the aforementioned sense. [not really a quote, just paraphrase] Specifically, for 0 C a fiber of a flat morphism 0 À \ Ä ], the dimension is given by dim \ dim ]. Lemma 1.7.1. For all subschemes ] ]ß 0 c] d œ c0 ] d. Proposition 1.7. Consider the fiber square ith 1 flat and 0 proper: 1 \ Ä \ 0 Æ Æ 0

] Ä 1 ] X2/81 is flat, 0 is proper, and for all α ^ \, 0 1 œ 1 0 α in ^ ]. Note. Ramin says What is a fiber square? Think about a product structure. What is a product? The product of to sets \ and ] is the Cartesian pairs BßC such that you have to projections onto B and onto C. The above is the situation hen B and C don't have any maps onto any other thing. No, if you did have a third map, then you'd ant the collection of pairs BßC hich actually map into the same thing in D. Set theoretically, for a diagram ^ Ä ] Æ Æ 0 \ Ä 1 ^, the fiber product is ^ \ ] given by ^ œ e BßC l1 B œ 0 C f. Suppose you have specv Ä Spec W and Spec X Ä Spec W and you ant to construct something Ä Spec V and Ä Spec X (diagram). First thing you do is reverse the arros so you get W Ä Vß W Ä X, and then the ring you construct is V Ä V Œ WX and X Ä V Œ WX. Then you consider the spectrum of these rings and that is the fiber product. Problem is, schemetheoretically you have to do this locally, so you have to make sure the data glues together. Proof. For \ and ] varieties ith α œ c\ d, assume surjective. Then 0 \ œ ] and 0 c\ d œ. c] d. We ant 0 c\ d œ. c] d. Take \ œ Spec P and ] œ Spec O ith OßP fields, and let ] œ Spec E ith E local Artinian, and E œ Spec F ith F œ EŒ O P. Then e are done by Lemma A.1.. We are no ready to prove the main theorem. Theorem 1.7. If 0 À \ Ä ] is a flat morphism (of relative dimension 8), define 0 À ^ 5] Ä ^ 5 8\ by 0 czd œ c0 Zd ( Z a variety). Then for α µ! in ^ 5], 0 α µ! in ^ 5 8\ Proof. Let α µ! in ^ O]. Then α œ cz! d cz d ith Z! œ ;:!, for

: À Z8 Ä and ; À ] Ä ] ith 0 À \ Ä ]. Take [ œ 0 Z a subscheme. Then 2 À [ Ä and : À \ Ä \ so that e can construct a fiber square So 0 \ Ò ] : Æ Æ 0 \ Ò ]. 0 0 α œ 0 c; :! d c; : d œ 0 ; c:! d c: d and then by the previous proposition, this equals : 0 c:! d c: d œ : 0 c:! d c: d œ : c2! d c2 d and so by Theorem 1.4 it suffices to sho c[ d œ 7 c[ d for [ œ - c d [. Need 2 œ 2l ith 2 T œ 7 c2 T d for T œ!ß. [ 1.8 An Exact Sequence See Proposition 1.8 and Example 1.9.. Lecture (February, 2009) Recall that if \ is a variety and Z \ is of codimension, then e defined a= b Zß\ (regular functions in the local ring), then j b Zß\ Î =. Furthermore, recall that a 5-cycle is a finite formal sum Z \ßdim Zœ5 8<cZd. Divisors Definition. Let \ be a variety ( 8-dimensional). Then a Weil divisor is an 8 -cycle on \. Furthermore, ^ \ œ abelian group of Weil divisors. 8 Definition. A C artier divisor on \ is defined by e Yαß0α f œ H, here Y α \ is open, and \ œ - αyα, and 0α are non-zero functions in V Yα œ V \ that satisfy (1) 0 α Î! on Yα, (2) 0α Î0 is a unit on Yα Y (a unit is nohere-vanishing, regular). Definition. The 's are called the local equation of. 0 H α

Definition. If H is a Cartier divisor of \, and Z \ is a subvariety of codimension ß then e can define an order function on the divisor, ord H ³ ord 0. Z This is ell-defined (independent of our choice of local equations) because they differ by a unit: ord 0α œ ord?0, here? is a unit in b Y Y ß\. Z α α Definition. The associated Weil divisor to H is chd œ ord HcZd. Z \ß codim Note. There are only finitely many Z \ of order ZH Á!. Definition. Let Div \ œ group of Cartier divisors. Let H œ Yαß0α ßI œ Yαß1α. Then e define H I œ Yαß0α1α. We also induce a homomorphism Div \ Ä ^ \ ith H È chd. Definition. For any 0 V \, e define a Principle Cartier Divisors div 0 by all local equations œ 0. Definition. To Cartier divisors H and are linearly equivalent if there exists an 0 V \ such that H œ H div 0. Definition. Pic \ œ Div \ Î µ, here µ is the above linear equivalence. Hence, H œ H div 0 Ö chd ch d µ! as 8 -cycle. This induces Pic \ Ä E \. 8 Definition. The support of a Cartier divisor HÎ\ denoted supp H, or khk is the unit of all subvarieties ^ \ such that if H œ Yαß0α then 0α is not a unit of b \ß^. Or 0α somehere on some ^ \ (that is, 0α has a zero or a pole on ^ \). Example. If \ is defined by D œ BC, then B œ D œ! is a Weil divisor, but not a Cartier divisor. This is because there is no ay to have local equations about the cycles define B œ D œ! exclusively. If D œ! Ö B œ D œ! or C œ D œ!. No matter hat local equation you choose, you ill alays get another line. $ # 2.1 - Line bundles as pseudo-divisors Definition. A pseudo-divisor is a triple Pß^ß= here P œ line bundle, ^ œ closed subset of \ (support), and s œ nohere vanishing section on \ ^. H 8

We say Pß^ß= œ P ß^ ß= if (i) ^ œ ^ ß (ii) b À P ĵ 5 P s.t. 5l \ ^ sends = È =. In other ords, they are equal if on the closed subsets e have isomorphic line bundles. (Recall isomorphism of line bundles!) Definition. For \ an algebraic scheme, E a Cartier divisor, a divisor H determines a line bundle b H by the sheaf of sections on the b\ -subsheaf generated by Î0 on Y (ith \ œ - Y ). Definition. A cartier divisor H is effective if the 0 œ = Hl Y for = H b\, here = H is the canonical section (i.e., if H has a canonical section that is regular, it is effective). Definition. A canonical divisor H determines a pseudo-divisor b H ßkHkß=. If Pß^ß= if khk œ ^ and b H z P. B H \ Lemma. If \ is a variety, any pseudo-divisor Pß^ß= is represented by a cartier divisor H such that (1) If ^ Á \, the operation is unique. (2) If ^ œ \, the operation is unique up to linear equivalence. Definition. If H is a pseudo-divisor on an 8-dimensional variety ith support khk, then the Weil divisor class is chd E khk. Definition. If E œ Pß^ß= and F œ P ß^ ß= are pseudo-divisors, then 8 E F œ PŒP ß^ ^ ß=Œ=, and, E œ P ß^ßÎ= here the tensor product is taken over the trivial bundle, PŒP ß^ß=Œ=. Definition. Let 0 À \ Ä \ ith H œ Pß^ß= Î\. Then 0 H œ 0 Pß0 D ß0 =. Notice 0 H H œ 0 PŒP ß^ ^ ß=Œ= œ 0 PŒP ß0 ^ ^ ß0 =Œ= œ 0 P Œ0 P ß0 ^ 0 ^ ß0 = Œ0 =. Intersecting ith divisors

Definition. If H is a pseudo-divisor on a scheme \ and Z is a 5- dimensional sub- variety of \, define HcZd or H Z in E5 khk Z to be 4 H, a pseudo-divisor on Z ith support khk Z. Then H Z is the Weil divisor class of 4 H œ c4 Hd. Note: If H is a Cartier divisor, and Z Î khk, then H retracts (pullback) to a Cartier divisor by 4 H. Definition. Let α œ 8ZcZd be a 5-cycle. Then the support of α, kαk is the union of subvarieties ith non-zero coefficients. Definition. Each H czd is a class in E5 khk kαk. We define the intersection class H α œ 8 H czd Z. Proposition. If H is a pseudo-divisor on \ and α is a 5-cycle, then (1) H α α œ H α H α, (2) H H α œ H α Hš () If 0 À \ Ä \ is a morphism, then there is 8 Z 1 À 0 khk kαk Ä khk 0 kαk ith 1 0 H α œ H 0 α. (4) Same as above but other direction. (5) If H is a pseudo-divisor ith b\ H is trivial, then H α œ! Lecture 4 (February 10, 2009) Theorem 2.4 Let HßH be Cartier divisors on an 8-dimensional variety \. Then 8 # H ch d œ H chd in E khk kh k, here ch d œ 8 cz d ^ \. codim Zœ 8 Proof. Assume H and are effective Cartier divisors that intersect properly (that is, there is no codimension subvariety of \ in their intersection, khk kh k). Recall the folloing fact: H

Let E be a local domain of dimension #. Take +ß+ E. Define / E +ßEÎ+ E as follos: +! Ä ker Ä EÎ+ E Ä EÎ+ E Ä coker Ä! Then let / E +ßEÎ+ E œ je coker je ker (length of cokernel minus length of kernel). Next, e ill need the lemmas: Lemma A.27. +ßEÎ+ E œ j E Î+ E j EÎ +E. e E E ht œ : SpecE Lemma A.28. e +ßEÎ+ E œ / + ßEÎ+E. E Note: For the above lemma, e ant H chd œ 7 ca d. EÎ H ch d œ 7 ca d to be Case 1 Assume H and are effective that intersect properly. Calculate the coefficient of [ in H ch d, ith dim E œ #. Then all primes ith ht correspond to a subvariety Z of codimension. Hence, H ch d œ ht œ : SpecE 8 cz d ÞÞÞ Here, 8 œ ord Z +. Since b Z ß\ œ E ß e further have 8 œ j E Î+ E. To continue, e first must compute E ith H ch d œ 8 H cz d, E H cz d œ 7 c[ d. Well, the coefficient of [ in H cz d is by the lemma. [ Z codim [ œ j EÎ+ E j EÎ + œ / +ßEÎ+ E EÎ EÎ Case 2 Assume H and are effective. We il reduce to the case of proper intersection. H E

Lemma. If H and H are Cartier divisors on \ and 1 À \ Ä \ is a proper birational morphism, ith 1 H œ F Gß 1 H œ F G ß kfk kg k 1 kf k kg k 1 kh k. khk ß We have four pairs on \, FßF ß FßG ß GßF ß and GßG. If the theorem holds true for each pair, then it is true in general. Definition. Let HßH be effective divisors on \. Define & HßH œ max ord ord e H H l codim Z œ f. Z Z \ Note that HßH intersect properly if and only if & HßH œ!. If & HßH!, then e blo up \ along khk khk. We get 1 À \ Ä \ ith exception divisor Iß so 1 H œ G I and 1 H œ G I. Then (a) G G œ g (b) if & HßH!, then & GßI & HßH and & G ßI & HßH. Case Assume H is effective. Let 1 be the denominator of the ideal sheaf of H. Then e can have a blo-up 1 À \ Ä Fl \ Ä \ ith 1 H œ G I 1 ith I an exceptional divisor. Then e can just consider the pair Gß H and Iß1 H and apply these to case 2. 1 Case 4 Let H and H be arbitrary divisors. Let 1 be the denominator of the ideal sheaf of H. We can have 1 À \ Ä Fl \ Ä \ ith 1 H œ G I. 1 Consider the pairs Gß1 H and Iß1 H. We can apply case to this and e're done! (of theorem) Corollary 2.4.1. Let H be a pseudodivisor on \. Let α ^ 5\ and α µ!. Then H α œ! in E khk ßE khk kαk. 5 5 Corollary 2.4.2. Let H and H be pseudo-divisors on \. Then for any α ^ 5 \. H H α œ H H α As cycles, H ch d Á H chd, but as classes, they are equal.

Example 2.4.1. Consider #. Consider the situation. First, e have H œ 1 G œ G I (principal), and H œ 1 G œ G I (principal), # # here by principal e mean b\ z b\ H and b\ z b\ H respect- ively. We have H ch d œ H cg d cid œ ðñò H cg d H cid : # #, here : is just a point (: ^! khk kg# k ) and E! khk E :. Notice! H cg# d ^! khk kg# k œ ^! l:l. W ecan also think of H cg# d E! kg # k and in that case : œ!, because G# œ. Thus, H cg# d œ :. On the other hand, H cid œ!. Why? Because ki k khk in ^ I. Hence, ðñò H cg d H cid œ : ^ I : #!. Notice that since I œ, the other one turns out to be ;. Hence, they are non-equal as cycles, but are as classes. Definition. Let H ßÞÞÞßH 8 be pseudodivisors on \. For any α ^ 5\, define H ÞÞÞ H α œ H H ÞÞÞ H α E kh k ÞÞÞ kh k kαk. 8 # 8 5 8 8 More generally, for any homogeneous polynomial of degree., ith integer coefficients. Then T X ßÞÞÞßX 8 T H ßÞÞÞßH α E kh k ÞÞÞ kh k kα k. 8 5 8!

If 8 œ 5ß ] œ kh k ÞÞÞ kh8k kαk is complete, define the intersection # to be H ÞÞÞ H α œ ' T H ßÞÞÞßH α. 8 \ D 8 Note. The integral means if α œ 8 ct d ith \ over 5 then ' α œ 8 c5 : 5 d. \ If Z is a pure 5 -dimensional subscheme of \, then Chern classes of line bundles 8 8 T H ßÞÞÞßH Z œ T H ßÞÞÞßH czd. Let P be a line bundle on a scheme \. We can define a homomorphism G P _ À ^ \ Ä E \ ith α œ 8 cz d È G P α 5 5 as follos. For any 5-dimensional subvariety Zß PlZ œ bz G here G is a cartier divisor on V Þ Then G P czd œ cgd E \. 5 Note that if P œ b \ H for a pseudo-divisor on \, then G P α œ H α. Proposition 2.5 (a) If α µ! on \, then G P α œ!. So e have G P À E \ Ä E \. 5 5 (b) If PßP are line bundles on \, α ^ \, then G P G P α œ G P G P α. (c) If 0 À \ Ä \ is a proper morphism, P is a line bundle on \, α is a 5- cycle on \, then 0 G 0 P α œ G P 0 α. (d) If 0 À \ Ä \ is flat of relative dimension 8, and P is a line bundle on \, ith α a 5 -cycle on \, then G 0 P 0 α œ 0 G P α. (e) If PßP are line bundles ith α ^ \ß then 5 5 G PŒP œ Definition. If P ßÞÞÞßP 8 are line bundles on \ßα E 5 \, and e have a homogeneous polynomial T X ßÞÞÞßX of degree., then 8

In particular, T G P ßÞÞÞßG P α E \. 8 5.. 5. G P E \. Definition. Let H be an effective divisor on \ and let À H Ä \ be the inclusion. Define the Gysin homomorphism À ^ 5\ Ä E5 H α È α œ H α. Proposition 2.6. (a) If α µ! on \, then α œ!. So e have À E 5\ Ä E5 H. (b) If α Z 5 \, then α œ G b H α. (c) If α ^5Hß, then here R œ b H. (d) If \ B α œ G R α, \ is purely 8-dimensional, then (e) If P is a line bundle on \, then for any α ^ \ÞJ?<>2/<ß 5 c\ d œ chd. G P α œ G P α C P G P α œ G P G P α. Lecture 5 (February 16, 2009) (1) T is projective if and only if T is proper. (2) b I ß proj EcB ßÞÞÞßB d.! / T T I Ä \, ith E \. α So = I α ³ T α. 8 8 8 (1) 0 À \ Ä \ß 0 = 0 I α œ = I 0 α. (a) (1) = 8 I œ! if!. (2) =! I α œ α.

\ H œ ˆ H ˆ b \ deg O. # \ \ \ \ \ \ g œ H œ b O œ b O œ... \ I œ ˆ # H H ˆ # deg O O G œ # # # degˆ # # HO H O G œ degˆ H H O # # # # # # # O G. # #