Yichao TIAN COURSE NOTES ON INTERSECTION THEORY

Yichao TIAN COURSE NOTES ON INTERSECTION THEORY

Yichao TIAN Mathematical Institute, University of Bonn, Endenicher Allee 60, Bonn, 53115, Germany. E-mail : tian@math.uni-bonn.de

COURSE NOTES ON INTERSECTION THEORY Yichao TIAN

CONTENTS 1. Chow groups.................................................................... 7 1.1. Cycles and rational equivalence............................................... 7 1.2. Proper push-forward of cycles................................................. 10 1.3. Geometric definition of rational equivalence................................... 13 1.4. Flat pull-back................................................................. 15 1.5. Affine bundles................................................................. 18 2. Intersection product on non-singular varieties.............................. 21 2.1. Moving Lemma and intersection products..................................... 21 2.2. Intersection Multiplicities..................................................... 25 2.3. Pullbacks for generically separable formulas................................... 27 3. Line bundles and intersection theory on surfaces.......................... 31 3.1. Cartier divisors............................................................... 31 3.2. Intersection with Cartier divisors............................................. 33 3.3. Intersection on surfaces....................................................... 37 4. vector bundles and Chern classes............................................ 41 4.1. Vector bundles and Chern classes............................................. 41 4.2. Chow groups of projective bundles............................................ 47 5. Gysin maps and excess intersection Formula............................... 53 5.1. Gysin homomorphisms of vector bundles...................................... 53 5.2. Excess intersections with regular embeddings................................. 55 5.3. Chow groups of the blow-up.................................................. 59 5.4. Application: The problem of five conics....................................... 61 Bibliography....................................................................... 65

CHAPTER 1 CHOW GROUPS 1.0.1. Notation. Let k be a field. All the schemes in this course are supposed to be separate of finite type over k. A variety is a separated, reduced and irreducible scheme over k. For a variety X, we denote by k(x) the field of rational functions on X. Let V X be a subvariety in a scheme X, we denote by O X,V the local ring of X at the generic point of V. 1.1. Cycles and rational equivalence Let X be a variety or a scheme (over k). For an integer d, a d-cycle on X is a finite formal sum n i [V i ] i where n i Z and V i X are subvarieties of dimension d. The set of d-cycles on X form a natural abelian group, which we denote by Z d (X). We put Z(X): = Z d (X), d=0 and call it the cycle group of X. A cycle i n i[v i ] is called effective if n i 0 for all i. Let Y X be a subscheme. Since Y is of finite type over k hence Noetherian, Y has only finitely many irreducible components, say Y 1,, Y s. Then each local ring O Y,Yi is Artinian. We define s [Y ]: = leng(o Y,ηi )[Y i ], and call it the cycle associated to Y. i=1 1.1.1. Order function. Let A be a one-dimensional Noetherian domain with fraction field K. One can define an order function ord A : K Z

8 CHAPTER 1. CHOW GROUPS as follows. For a A\{0}, A/(a) is a finite length A-module, and we put Then it is easy to see that For x = a b K, we put ord A (a) = leng(a/(a)). leng(a/(ab)) = leng(a/(a)) + leng(a/(b)), a, b A. ord A (x): = ord A (a) ord A (b), which is independent of the writing x = a b. It is easily seen that ord A is multiplicative: ord A (xy) = ord A (x) + ord A (y), x, y K. Note that we do not require that A to be neither local nor normal. If this is the case, A is a discrete valuation ring, and there exists a uniformizer π A such that any element f k(x) writes uniquely as f = π n u with u A and ord A (f) = n. Let X be a variety of dimension n, and V X be a codimension 1 subvariety. The ring O X,V is a one-dimensional local domain. For any f k(x), we put ord V (f) = ord OX,V (f), and define the divisor associated to f as div(f) = V X ord V (X)[V ], where the sum runs through the closed subvarieties of codimension 1 of X. If ord V (f) = n > 0 (resp. ord V (f) = n < 0), then we say that f has a zero (resp. pole) of order n at V. By the additivity of the functions ord V, the map f div(f) defines a homomorphism of groups div: k(x) Z n 1 (X). If there is risk of confusion, we use the notation div X to emphasize the dependence on X. Example 1.1.2. Let C be the projective plane curve with homogeneous equation Y 2 Z = X 3. Let x = X Z and y = Y Z. Then one has k(c) = k(x, y)/(y2 x 3 ), and div C (x) = 2[P ] 2[ ] and div C (y) = 3[P ] 3[ ], where P is the point (0, 0, 1) C and = (0, 1, 0). If U denotes the affine curve C\{ }, then div U (x) = 2[P ] and div U (y) = 3[P ]. Definition 1.1.3. Let X be a scheme over k. A d-cycle α Z d (X) is rationally equivalent to 0, denoted as α 0, if there exist a finite number of (d + 1)-dimensional closed subvarieties W i X and f i k(w i ) such that α = div(f i ). i Two d-cycles α, β A d (X) are called rationally equivalent, written as α β, if α β is rationally equivalent to 0. The d-cycles rationally equivalent to 0 form a subgroup Rat d (X) Z d (X). We put A d (X): = Z d (X)/ Rat d (X),

1.1. CYCLES AND RATIONAL EQUIVALENCE 9 and call it the Chow group of d-dimensional cycles on X. We put also A(X) = d A d (X). Example 1.1.4. (a) One has A 0 (P 1 ) = Z[ ] and A 1 (P 1 ) = Z[P 1 ]. (b) One has A 0 (A 1 ) = 0 and A 1 (A 1 ) = Z[A 1 ]. We have the following basic properties of Chow groups: Lemma 1.1.5. Let X be a scheme over k. (a) If X red X is the closed reduced subscheme, then A d (X) = A d (X red ). (b) If X is a disjoint union of schemes X 1,, X s, then Z d (X) = Z d (X i ) and A d (X) = A d (X i ). (c) If X 1, X 2 are closed subschemes of X, then there are exact sequences A d (X 1 X 2 ) A d (X 1 ) A d (X 2 ) A d (X 1 X 2 ) 0, where the first arrow is α (α, α) and the second is (β, γ) β γ. (d) Let Y X a closed subscheme, and U = X\Y. Then one has an exact sequence A d (Y ) α A d (X) β A d (U) 0, where α is the natural map, and β is induced by η = i n i[v i ] η U = i n i[v i U]. Proof. The statements (a) and (b) are evident. To prove (c), note that the natural mapz d (X 1 ) Z d (X 2 ) Z d (X 1 X 2 ) is surjective, hence so is A d (X 1 ) A d (X 2 ) A d (X 1 X 2 ). To show that the exactness in the middle, let α Z d (X 1 ) and β Z d (X 2 ) be such that α β Rat d (X 1 X 2 ). Then by definition, there exists a finite number of (d + 1)-dimensional subvarieties W i X 1 X 2 and f i k(w i ) such that α β = i I div(f i ). Since each W i is irreducible and reduced, one has either W i X 1 or W i X 2. There exists thus a partition I = I 1 I 2 such that i I j if and only if W i X j. Note that i I j div(f i ) Rat d (X j ). Therefore, up to replacing α by α i I 1 div(f i ) and β by β i I 2 div(f i ), we may assume that α = β. This means that (α, β) lies in the image of Z d (X 1 X 2 ) Z d (X 1 ) Z d (X 2 ). For (d), one has obviously an exact sequence (1.1.5.1) 0 Z d (Y ) Z d (X) U Z d (U) 0, from which the surjectivity of β : A d (X) A d (U) follows. Now let η Z d (X) be a d-cycle such that η U 0. One has to show that η is rationally equivalent to d-cycle in Y. Indeed, the assumption on η implies that there exist (d + 1)-dimensional closed subvarieties W i U and f i k(w i ) such that η U = i div Wi (f i )

10 CHAPTER 1. CHOW GROUPS Let W i denote the Zariski closure of W i in X. Then one has η U = i div W i (f i ) U. By the exact sequence (1.1.5.1), it follows that η i div Wi (f i ) Z d (Y ). Remark 1.1.6. In general, the morphism α in Lemma 1.1.5(d) is not injective. 1.2. Proper push-forward of cycles Let f : X Y be a proper morphism of schemes. If V X is a closed subvariety of dimension d, its image W = f(v ) is closed in Y. There is an induced embedding k(w ) k(v ). Note that k(v )/k(w ) must be a finite extension of field, since they are both finitely generated extensions over k of the same transcendence degree. We put { 0 if dim(w ) < dim(v ); f [V ] = [k(v ) : k(w )][W ] if dim(w ) = dim(v ). This extends by linearity to a homomorphism of d-cycles f : Z d (X) Z d (Y ). From this definition, it is clear that (gf) = g f for any proper morphism g : Y Z. Theorem 1.2.1. The map f : Z d (X) Z d (Y ) preserves linear equivalence, and it induces thus a map f : A d (X) A d (Y ). Proof. If α = div(r) is a d-cycle given by a rational function r defined over a closed subvariety V X. One has to show that f (div(r)) is also rationally equivalent to 0. Up to replacing X by V and Y by f(v ), one reduces to show the following Proposition 1.2.2. Let f : X Y be a proper, surjective morphism of varieties, and r k(x). Then { 0 if dim(y ) < dim(x); f (div(r)) = div(n(r)) if dim(y ) = dim(x), where N(r) k(y ) is the norm of r. Before proving Proposition 1.2.2, one deduces some corollary from it. Let X be a proper scheme, i.e. the structural morphism p : X S = Spec(k) is proper. For any zero-cycle α = P n P [P ] Z 0 (X), we define the degree of α as deg(α) = α = n P [k(p ) : k]; X P or equivalently, if we identify A 0 (S) = Z[S] with Z via the basis [P ], we have called the degree of α. deg(α) = p (α),

1.2. PROPER PUSH-FORWARD OF CYCLES 11 Corollary 1.2.3. (a) If X is a proper scheme, then the degree map induces a homomorphism : A 0 (X) Z. X (b) If f : X Y is a morphism of proper schemes, then Y f (α) = X α for any α A 0 (X). Proof. (a) By Theorem 1.2.1, p : Z 0 (X) Z 0 (S) induces a map p : A 0 (X) A 0 (S) = Z. (b) It follows from p = q f, where q : Y S is the structural map of Y. We now turn to the proof of Proposition 1.2.2. Let R be a domain with fractional field K. If M is a finitely generated torsion free R-module and ϕ : M M is a R-endomorphism, then ϕ R K is an endomorphism of the finite dimensional K-vector space M R K. We define det(ϕ) K as the determinant of ϕ R K. Lemma 1.2.4. Let R be a 1-dimensional local Noetherian domain which is the localization of a finitely generated k-algebra, and ϕ : M M be an endomorphism of a finitely generated torsion free R-module. Assume that det(ϕ) 0. Then ϕ is injective and we have leng(m/ϕ(m)) = ord R (det(ϕ)), where ord R : K Z is the order function defined in Subsection 1.1.1. In particular, if M has rank n, then one has, for any r R, leng(m/rm) = nleng(r/rr) = n ord R (r). Proof. The endomorphism ϕ R K is an isomorphism if and only if det(ϕ) 0. The injectivity of ϕ follows immediately from the fact that M is torsion free. First, we reduce the problem to the case when R is a discrete valuation ring. Let R be the integral closure of R in K. Then R is a Dedekind domain, and the quotient R /R is a R-module of finite length. Let M be the quotient of M R R by its torsion submodule. Then one has a natural inclusion M M, and the quotient M /M is a R-module of finite length. Moreover, ϕ induces an endomorphism ϕ : M M of R -modules, and one has a commutative diagram By Snake Lemma, one obtains 0 M M ϕ ϕ M /M 0 M M M /M 0. 0 Ker( ϕ ) Coker(ϕ) Coker(ϕ ) Coker( ϕ ) 0. By the additivity of the length function, one gets leng(coker(ϕ)) leng(coker(ϕ )) = leng(ker( ϕ )) leng(coker( ϕ )). ϕ 0

12 CHAPTER 1. CHOW GROUPS On the other hand, it follows from the exact sequence that 0 Ker( ϕ ) M /M M /M Coker( ϕ ) 0 leng(ker( ϕ )) leng(coker( ϕ )) = leng(m /M) leng(m /M) = 0. Therefore, one gets leng(coker(ϕ)) = leng(coker(ϕ )). Note that ϕ R K = ϕ R K. Hence, one is reduced to proving that det(ϕ ) = leng(coker(ϕ )). Up to replacing R by R and M by M, one may assume that R is a Dedekind domain with finitely many maximal ideals, hence R is a principal ideal domain. By the structural theorem of finitely generated modules over a PID, there exists a basis (e 1,, e n ) of M over R and a 1,, a n R such that (a 1 e 1,, a n e n ) is a basis of ϕ(m). Therefore, one has n leng(m/ϕ(m)) = leng(r/a i R). On the other hand, since (ϕ(e 1 ),, ϕ(e n )) is another basis of ϕ(m), there exists a matrix B GL n (R) such that i=1 (ϕ(e 1 ),, ϕ(e n )) = (a 1 e 1,, a n e n )B, i.e., ϕ has matrix diag(a 1,, a n )B under the basis (e 1,, e n ). Hence, one gets det(ϕ) = a 1 a n u with u = det(b) R, and n ord R (det(ϕ)) = ord R ( a i ) = i=1 n leng(r/a i R) = leng(m/ϕ(m)). i=1 Proof of Proposition 1.2.2. (a) Assume dim(y ) = dim(x). Let X X = Spec(f (O X )) Y denote the Stein factorization of f. We fix a subvariety W of Y of codimension 1. Put A = O Y,W, and B = f (O X ) Y,W. Then A is a Noetherian local domain of dimension 1, and B is a finite A-algebra such that the maximal ideals m i of B correspond to the subvarieties V i X mapping onto W. Then to prove the Proposition in this case, it suffices to show that ord Vi (r)[k(v i ) : k(w )] = ord W (N(r)). i Since both sides above are multiplicative in r, it is enough to prove this when r B\{0}. Consider the homomorphism B r B. By Lemma 1.2.4, we have ord W (N(r)) = leng A (B/rB),

1.3. GEOMETRIC DEFINITION OF RATIONAL EQUIVALENCE 13 where leng A means the length as A-module. But B/rB is a Artinian ring, and we have leng(b/rb) = i = i leng A (B mi /rb mi ) = leng Bmi (B mi /rb mi )[k(m i ) : k(m A )] ord Vi (r)[k(v i ) : k(w )]. This finishes the proof of Proposition 1.2.2 when dim(x) = dim(y ). (b) Assume dim(y ) < dim(x). Consider first the case Y = Spec(K) is the spectrum of a field K, and X = P 1 K. The field of rational functions on X is K(t). We may assume that r is an irreducible polynomial in t of degree d. Then ideal (r) K[t] corresponds to a point P of P 1 K of degree d, and we have div(r) = [P ] d[ ]. Note that K(P ) is a finite extension of K of degree d. One obtains f (div(r)) = d[y ] d[y ] = 0. In the general case, one may assume that dim(y ) = dim(x) 1, otherwise the statement is trivial. f (div(r)) is a multiple of [Y ] with coefficients ord V (r)[k(v ) : K], V with K = k(y ), where V runs through all the codimension 1 subvarieties of X mapping surjectively to Y. Put X K = X Y Spec(K). So one may assume that X is a curve over Y = Spec(K). Let g : X X denote the normalization. Then the rational function r defines a finite dominant map h : X P 1 K. One has thus a commutative diagram: X g X h P 1 K p f Spec(K) If r denotes the rational function of X defined by the isomorphism K( X) = K(X), one has f (div(r)) = f g (div( r)) = p h div( r) However, one has h (div( r)) = div(n(r)) by part (a). dim(y ) < dim(x) follows from the case of P 1. Hence, Proposition 1.2.2 when 1.3. Geometric definition of rational equivalence Let X be a scheme. We introduce a boundary map : Z(X P 1 ) Z(X) as follows: Let W X P 1 be a closed subvariety. If pr 2 : W P 1 is not dominant, then pr 2 (W ) must be a closed point of P 1 and we put ([W ]) = 0. If pr 2 : W P 1 is dominant, then

14 CHAPTER 1. CHOW GROUPS put W 0 = pr 1 (pr 1 2 ({0})) and W = pr 1 (pr 1 2 ({ })) (scheme theoretic preimage). We define ([W ]) = [W 0 ] [W ]. Proposition 1.3.1. A cycle α Z d (X) is rationally equivalent to 0 if and only if it is a linear combination of cycles of the form (W ), for some subvariety W X P 1 k of dimension d + 1 dominating over P 1. Proof. First, we prove that (W ) is rationally equivalent to 0. Indeed, the second projection pr 2 : W P 1 gives rise to a rational function r k(w ). Then we have div(r) = pr 1 2 (W 0) pr 1 2 (W ), and (W ) = pr 1, (div(r)). It follows from Proposition 1.2.2 that (W ) is rationally equivalent to 0. Conversely, if V is a subvariety of X of dimension d + 1 and r k(v ), we prove that div(r) is has the form (W ). Indeed, such a rational function gives a rational morphism from V to P 1, i.e. there exists an open dense subset U V such that r : U P 1 is dominant. We take W to be the closure of the graph of r in X P 1. Then the projection to X maps W birationally and properly onto V. Then we have div(r) = (W ). Example 1.3.2. Let C, D P N be two distinct hypersurfaces defined by two homogeneous polynomials F (x), G(x) k[x 0, x 1,, x N ] of the same degree. We define W = {(x, t) P N P 1 t 0 F (x) + t 1 G(x) = 0}, where t = (t 0, t 1 ) is the homogeneous coordinate on P 1. Then W P 1 is dominant it is clear that W 0 = C and W = D, and hence C D. In particular, if S d P N is any hypersurface of degree d, then [S d ] = d[h] with H P N a hyperplane. 1.3.3. Hilbert polynomial. For any coherent sheaf F on a proper scheme Y, recall that the Euler-Poincaré characteristic of F is defined as χ(f): = ( 1) i dim k H i (Y, F). i=0 Assume Y P N is projective. Let O Y (1) be the canonical ample line bundle, and O Y (n) the n-th twist. We put F(n) = F OY O Y (n). The function n χ(f(n)) is a polynomial in n, and call it the Hilbert polynomial of F. If F = O Z for some closed subvariety Z Y, we also call χ(o Z (n)) the Hilbert polynomial of Z. Proposition 1.3.4. Assume X is proper. Let W X P 1 be a closed subvariety dominating P 1, and p, q P 1 (k) be two points lying in the image of W. Let W p and W q denote respectively the fibres of W at p and q. Then we have χ(o Wp ) = χ(o Wq ). Moreover, if X is projective, then the Hilbert polynomial of W p and W q are the same, i.e. we have χ(o Wp (n)) = χ(o Wq (n)), n Z

1.4. FLAT PULL-BACK 15 Proof. For any some variety Z X P 1, denote by O Z (0, b) the pull-back to Z of pr 2 O P 1(b). Let (t 0, t 1 ) denote coordinates on P 1 such that p (resp. q) is defined as t 0 = 0 (resp. t 1 = 0). Since W is a variety dominating P 1, it is flat over P 1. One has two sequences 0 O W (0, 1) t 0 O W O Wp 0 0 O W (0, 1) t 1 O W O Wq 0, and these sequences are exact because W is a variety dominating over P 1, hence flat over P 1 around p, q. Now the equality follows from the additivity of χ. Now assume that X is projective. Twisting the sequences by pr 1 O X(n) and using the additivity of Euler-Poincaré characteristics, one gets χ(o Wp (n)) = χ(o Wq (n)), n Z. Remark 1.3.5. The previous Proposition is a special case of the following general fact: Let f : X S be a proper morphism of schemes, and F be a coherent O X -sheaf flat over S. For any s S, put F s = F f 1 O S k(s). Then χ(f s ) is a locally constant function in s S. Assume X is proper. By the additivity of Euler-Poincaré characteristic, one can extend χ by linearity to Z(X). By Proposition 1.3.4, χ induces a function χ : A(X) Z. If X is projective, one can even attach a Hilbert polynomial P α (x) to a cycle class α A(X): if α = V n V [V ], P α (x) Q[x] is the polynomial such that P α (x) := V n V χ(o V (x)), x Z. 1.4. Flat pull-back Let f : X Y be a flat morphism of schemes. Assume that f has relative dimension n, i.e. for any subvariety V of Y and all irreducible components V of f 1 (V ), one has dim(v ) = dim(v ) + n. For any closed subvariety V Y of dimension d, set f [V ] = [f 1 (V )]. This extends by linearity to a pull-back morphism f : Z d (Y ) Z d+n (X). Lemma 1.4.1. For any closed subscheme Z Y, one has f [Z] = [f 1 (Z)].

16 CHAPTER 1. CHOW GROUPS Proof. By definition, one has f [Z] = f V Z leng(o Z,V )f [V ] = V leng(o Z,V )[f 1 (V )] and [f 1 (Z)] = W f 1 (Z) leng(o f 1 (Z),W )[W ], where V (resp. W ) runs through the irreducible (reduced) components of Z (resp. of f 1 Z). For each irreducible component W of f 1 (Z), the closure of the image f(w ), say V, must be an irreducible component of Z. Let A = O Z,V and B = O f 1 (Z),W. Then both A and B are local Artinian rings, so that A admits a decreasing filtration by A-submodules M 0 = A M 1 M n = 0 such that each M i /M i+1 is isomorphic to A/m A. By the flatness of B, one has leng B (B) = n leng B (B/m A B) = leng A (A)leng B (B/m A B). But the left hand side is the coefficient of [W ] in [f 1 (Z)], and the right hand side is that of [W ] in f [Z]. The Lemma follows immediately. Proposition 1.4.2. Given a cartesian diagram of schemes X g X f Y with f proper and g flat of relative dimension n, then, for any α Z d (X), the equality in Z d+n (Y ). g Y g f (α) = f g (α) Proof. We may assume that α = [V ] for some closed subvariety of X of dimension d. Put W = f(v ), V = g 1 (V ) and W = g 1 (W ). Then every irreducible component of V (resp. of W ) has dimension d + n (resp. dim(w ) + n). So if dim(w ) < dim(v ), then the desired equality is trivial. Assume thus dim(w ) = d. Up to replacing Y by W and X by V, one may assume that f : X Y is a proper surjective morphism of varieties and α = [X]. Since the problem is local for Y, after taking the base change Spec(k(Y )) Y, one may assume that Y = Spec(K) and X = Spec(L) with L/K finite. Then one has On the other hand, we has g f ([X]) = [L : K]g ([Y ]) = [L : K][Y ]. f g ([X]) = f ([Y K L]) = [L : K][Y ]. f

1.4. FLAT PULL-BACK 17 Theorem 1.4.3. Let f : X Y be a flat morphism of relative dimension n. Then f preserves rational equivalence, hence it induces a homomorphism of Chow groups: f : A d (Y ) A d+n (X) Proof. Let W Y be a closed subvariety of dimension d + 1, and r k(w ). We have to show that f div(r) is rationally equivalent to 0. Up to replacing Y by W f(x), we may assume that Y = W is a variety and f is surjective. Let X i be the irreducible components of X, and [X] = i n i [X i ], with n i = leng(o X,Xi ). Then for each i, the restriction f Xi induces an injection k(y ) k(x i ). Let r i k(x i ) denote the image of r. We claim that f div(r) = i n i [div(r i )]. By definition, we have f div(r) = V ord V (r)f [V ] = V ord V (r)[f 1 (V )]. Fix a subvariety Z X of codimension 1. Let V be the closure of f(z). Put A = O Y,V and B = O X,Z. Then A is a Noetherian local domain of dimension 1 with fraction field k(y ), and B is a local ring of dimension 1 flat over A. The minimal primes p i of B are in bijection with the irreducible components X i such that Z X i. Then the coefficient of [Z] in f div(r) is ord V (r)leng(b/m A B), and that in i n i[div(r i )] is p i leng(b pi )ord Z (r i ). We may assume that r m A. To finish the proof, it is enough to show that leng(a/ra)leng(b/m A B) = p i leng(b pi )leng(b/p i + rb). Since B is flat over A, one sees easily that leng(a/ra)leng(b/m A B) = leng(b/rb). Now the Theorem follows immediately from the following Lemma. Lemma 1.4.4. Let B be a Noetherian local ring of dimension 1 with minimal ideals p i, and M is a finitely generated B-module. Let x m B which is M-regular, i.e. M x M is injective. Then one has leng(m/xm) = p i leng(m pi )leng(b/(p i + xb)). Proof. See [Fu98, Lemma A.2.7.]

18 CHAPTER 1. CHOW GROUPS 1.5. Affine bundles A scheme p : E X over X, is called an affine bundle of rank n, if there exists an open covering U α of X such that p 1 (U α ) = U α A n. Proposition 1.5.1. Let p : E X be an affine bundle of rank n. Then the pull-back morphism p : A d (X) A d+n (E) is surjective. Proof. By Noetherian induction on X, we may assume that the affine bundle is trivial, i.e. E = X A n and X is an affine variety. By induction on n, we reduce to the case n = 1. One has to prove that every d + 1-dimensional subvariety V X A 1 is rationally equivalent to a cycle of the form p α. We may assume that V dominates X. Let A be the coordinate ring of X with fraction field K. Let q A[t] be the prime ideal corresponding to the subvariety V X A 1. If dim(x) = d, then V = E and hence [V ] = p [X]. Otherwise, one has dim(x) = d + 1. Since V dominates X, the prime ideal qk[t] K[t] is generated by a polynomial r K[t]. Then [V ] div(r) = i n i [V i ] for some subvarieties V i X A 1 of dimension d + 1 not dominating X. Putting W i = p(v i ), then one has V i = p 1 (W i ). The Proposition follows immediately. Corollary 1.5.2. Let U be an open dense subset of A n. Then A d (U) = 0 for any d n, and A n (U) is a free abelian group generated by [U]. Proof. For U = A n, this follows immediately from Proposition 1.5.1. The general case follows from Lemma 1.1.5(d). Corollary 1.5.3. Let X be a scheme with a cellular decomposition: X = X n X n 1 X 0 X 1 = by closed subschemes, such that each X i X i 1 is a disjoint union of schemes U i,j isomorphic to A n i,j. Let V i,j denote the closure in X of U i,j. Then the Chow group A(X) is generated by [V i,j ]. Proof. We proceed by induction on m with m n to show that A(X m ) is generated by [V i,j ] for i m. This follows immediately from Corollary 1.5.2 and the exact sequence: A(X m 1 ) A(X m ) A(X m X m 1 ) 0. Example 1.5.4. Consider the projective space P n. One has a cellular decomposition: P n P n 1 P 1 P 0 = {(0,, 0, 1)}, where P n r is the closed subvariety of P n defined by vanishing of the first r-coordinates. Denote by L r A n r (P n ) the class of P n r. By Corollary 1.5.3, A n r (P n ) is generated by

1.5. AFFINE BUNDLES 19 L r. We claim that A n r (P n ) is actually a free abelian group with generator L r. Indeed, it suffices to see that L r is not a torsion class in A n r (X). Recall that the Hilbert polynomial of a class of A(P n ) is well defined, and that of L r is ( ) x + n r P L r(x) = χ(o P n r(x)) =. n r Hence, L r is not torsion, and we conclude that A n r (P n ) = Z L r. Now let X P n be an arbitrary closed subscheme of pure dimension n r. The Hilbert polynomial of X has the form x n r P X (x) = c 0 (n r)! + lower terms, with c 0 Z\{0}. We put deg(x) = c 0 and call it the degree of X. Then it is easy to see that [X] = deg(x)l d in A n d (P n ). Exercise 1.5.5. Let S P N be a hypersurface defined by a homogeneous equation of degree d. Then the degree of S in the sense above is d. Example 1.5.6. Let G be a reductive group over k, T B G be a maximal torus and Boreal subgroup defined over k. Let W = N G (T )/T be the Weyl group. Consider the flag variety. One has the Bruhat decomposition: G/B = BẇB/B, w W where ẇ is a representative of w W. Note that, for each w W, we have BẇB/B = B/B ẇbẇ 1 = A l(w), where l(w) is the length of w W. Denote by BẇB/B the closure of BẇB/B in G/B. Then we have BẇB/B BẇB/B = Bw B/B. w <w In particular, G/B = w W BẇB/B is a cellular decomposition of G/B. Let σ w denote the class of BẇB/B in A(G/B). Then by Corollary 1.5.3, A(G/B) is generated by {σ w : w G/B}.

CHAPTER 2 INTERSECTION PRODUCT ON NON-SINGULAR VARIETIES We are going to define the intersection product on the Chow group of a variety X over k. There are two approaches to the definition of intersection product. The first approach uses moving lemma, and it applies only to smooth quasi-projective varieties over k. The second approach uses the technique of deformation to the normal cone, and this is the approach adapted in [Fu98]. In this chapter, we will discuss the first approach via moving Lemma by following [EH13]. In this chapter, we assume k is algebraically closed. 2.1. Moving Lemma and intersection products Definition 2.1.1. Let X be a variety. (a) Let A, B be subschemes of X. We say A and B are dimensionally transverse if for every irreducible component C of A B, one has codim X (C) = codim X (A) + codim X (B). (b) Subvarieties A and B are transverse at a point p X if X, A and B are smooth at p and their tangent spaces satisfy T p (X) = T p (A) + T p (B). (c) Subvarieties A, B are generically transverse if every irreducible component of A B contains a point p at which A and B are transverse. Lemma 2.1.2. Subvarieties A, B of X are generically transverse if and only if they are dimensionally transverse and each irreducible component C of A B is reduced and contains a closed point of p such that C and X are smooth at p. Proof. First, we suppose that A and B are generically transverse. By definition, every irreducible component C of A B contains a point p X such that A, B and X are all smooth at p and T p X = T p A + T p B. One has to show that C is smooth at p and codim(c) = codim(a) + codim(b). Indeed, Let R = O X,p and I R and J R be the ideals given by A and B. By assumption, R, R/I and R/J are all regular local rings of

22 CHAPTER 2. INTERSECTION PRODUCT ON NON-SINGULAR VARIETIES dimension n = dim(x), a = dim(a) and b = dim(b). Let m denote the maximal ideal of R. The condition T p X = T p A + T p B is equivalent to saying that the natural map m/m 2 m/(i + m 2 ) m/(j + m 2 ) is injective, i.e. if Ī and J denote respectively the image of I and J in m/m 2, then one has Ī J = 0. It follows that the image of I + J in m/m 2 has dimension n a + n b, and R/(I + J) = O C,p is regular of dimension n (2n a b) = a + b n. Let C be an irreducible component of A B such that codim(c) = codim(a)+codim(b) and C contains a smooth point p C. Let R, I, J be as above. Then one has dim(a) = dim(r/i) dim k m/(i + m 2 ) = n dim k (Ī), i.e. dim k Ī codim(a), and the equality holds if and only if A is smooth at p. Similarly, we have dim k J codim(b). But C is smooth at p, it follows that codim(c) = dim k (I + J) dim k (Ī) + dim k( J) codim(a) + codim(b). As codim(c) = codim(a) + codim(b), hence all the equalities above hold, that is, A and B are smooth at p, and dim k (I + J) = dim k (Ī) + dim k( J), which is equivalent to saying that T p X = T p A + T p B. It is natural to generalize the notion of dimensional transverseness from subvarieties to cycles. Two cycles α = i m i[a i ] and β = j n j[b j ] are called dimensionally (resp. generically) transverse if A i, B j are dimensionally (resp. generically) transverse. If A = i m i[a i ] and B = j n j[b j ] are two cycles on X intersecting generically transversely, then we put A B = m i n j [A i B j ]. i,j Theorem 2.1.3 (Moving Lemma). Let X be a smooth quasi-projective variety. (a) Let α A(X) and B Z(X). Then there exists a cycle A Z(X) which represents α and intersects B generically transversely. (b) If A and B are cycles intersecting generically transversely, then the class [A B] A(X) depends only on the class [A], [B] A(X). Remark 2.1.4. Even if α and β are classes of effective cycles, it is not always possible to find effective representatives A, B in the Moving Lemma intersecting transversally. Given the moving Lemma, one can define the intersection product on the Chow group of a smooth quasi-projective variety. Theorem 2.1.5. Let X be a smooth quasi-projective variety. There exists a unique bilinear product on A(X) such that [A] [B] = [A B] whenever A and B are subvarieties that are generically transverse. Under this product structure, A(X) becomes an associative, commutative ring.

2.1. MOVING LEMMA AND INTERSECTION PRODUCTS 23 Assuming the Moving Lemma, the proof of this Theorem is straightforward, and we will leave it as an exercise. Let n = dim(x). For each integer i 0, we put A i (X) = A n i (X). Then it is easy to see that A i (X) A j (X) A i+j (X). Therefore, A(X) = i Z Ai (X) is a commutative graded ring. Example 2.1.6. Let ξ A 1 (P n ) denote the class of a hyperplane. Then ξ d is the class of a (n d)-dimensional linear subspace. We have seen that A d (P n ) is a free abelian group generated by ξ d. Thus, we have an isomorphism of commutative graded rings: A(P n ) = Z[ξ]/ξ n+1. Recall that for each subvariety X P n of codimension d (or more generally, for a cycle α A d (P n )), we have defined a degree deg(x). It has the following geometric interpretation: deg(x) is the number of intersection points of X with a general linear subspace in P n of dimension d. Let H 1,, H n be n hypersurfaces of degree d 1,, d n respectively. If H 1 H n consists of some reduce points, then #(H 1 H n ) = deg([h 1 ] [H n ]) = i Example 2.1.7. Using the Chow ring of P n, one can show the following statement: Any morphism from P n to a quasi-projective variety of dimension strictly less than n is constant. Indeed, suppose ϕ : P n X is such a map. We may assume that ϕ is surjective. Suppose that ϕ is not constant, i.e. dim(x) > 0. Then the preimage of a general hyperplane section of X will be disjoint from the preimage of a general point of X. The preimage of a general hyperplane section of X has dimension n 1 and that of a general point on X has dimension > 0. But two such subvarieties of P n must meet, which is a contradiction. Example 2.1.8. Consider the Veronese embedding v n,d : P n P N, [x 0,, x n ] [, X I, ], where N = ( ) n+d d 1 and X I runs over all monomials of degree d in n + 1 variables. Denote by Φ n,d its image. Then deg(φ n,d ) is the number of points of the intersection of Φ n,d with n general hyperplanes. Since the inverse image of a hyperplane in P N via v n,d is a hypersurface of degree d, thus deg(φ n,d ) equals also to the number of n general hypersurfaces of degree d, which is d n by Bezout. Example 2.1.9. A product of projective spaces has also an evident cellular decomposition. It follows that A(P n 1 P nr ) = Z[t 1,, t r ]/(t n 1+1 1,, t nr+1 r ). This can be viewed as a Kunneth formula for Chows groups of projective spaces. However, note that Kunneth formula is in general FALSE for Chow groups. We focus on the case of r = 2. d i.

24 CHAPTER 2. INTERSECTION PRODUCT ON NON-SINGULAR VARIETIES As shown by the following examples, the smoothness of X is essential for the moving Lemma to hold. Example 2.1.10. Let C P 2 P 3 be a smooth conic and p / P 2. Let X = pc P 3 be the cone with vertex p. Let L X be a line passing through p. We claim that every cycle on X rationally equivalent to L has support containing the singular point p, and hence the conclusion of the first part of the Moving Lemma fails for X. Via the natural embedding X P 3, one can define the degree for cycles on X. It is clear that deg(l) = 1 and deg(c) = 2. On the other hand, if D X is a curve not containing p, then the natural projection from p defines a finite flat map D C. We claim that deg(d) = deg(f) deg(c). Indeed, choose a line H P 2 cutting transversally with C, and let H = ph be the unique plane containing p and H. Then we have deg(d) = #(H D) = deg(f)#(h C) = deg(f) deg(c). Hence, any cycle of dimension 1 on X with support disjoint from p has even degree. Since the degree is invariant under rational equivalence, the claim follows. Example 2.1.11. Let Q P 3 be a smooth quadratic surface and let X = pq be the cone in P 4 with vertex point p / P 3. It is well known that the quadratic surface Q contains two rulings of lines {M t } and {N t } parametrized by t P 1, and they are the only lines contained in Q. For instance, if Q is the surface with affine equation x 2 + y 2 z 2 = 1, then the lines are parametrized by the circle x 2 + y 2 = 1 with equations { x = ±az + b with a 2 + b 2 = 1. y = bz + a Correspondingly, the cone has two families of planes {Λ t = pm t } and {Γ t = pn t }. Let L X be a line not passing through p. Under the projection from p, L maps a line on Q. Hence, its image must lies in a plane Λ t or Γ t. Note that lines M t and M t on Q in the same ruling are disjoint if t t, and lines M t and N t from different ruling intersect at one point. It follows that, if M is a general line lying on Λ t, then M is disjoint from Λ t for t t and cuts with Γ t at one point. Therefore, if there exists a good theory of intersection product on X satisfying the same condition as in Theorem 2.1.5, one would have [M] [Λ t ] = 0, [M] [Γ t ] = [q], for some point q X. Similarly, if N is a general line on Γ t, one would have [N] [Λ t ] = [r], [N] [Γ t ] = 0. Note also that [q], [r] A 0 (X) are necessarily non-zero, since their degrees are non-zero. However, note that all lines on X are rationally equivalent, because one can move a general line M on Λ t to a line passing through p, and them move it to a general line on the other ruling Γ t.

2.2. INTERSECTION MULTIPLICITIES 25 2.2. Intersection Multiplicities Let X be a smooth quasi-projective variety. We have defined the intersection product on A(X) using Moving Lemma, so that if A, B are two subvarieties are generically transversal, then [A][B] = [A B]. We want to extend this formula to the case when A, B are just dimensionally transversal. Definition 2.2.1. Let A and B be two subschemes of X which are dimensionally transverse, and let Z be an irreducible component of A B. We define the intersection multiplicity of A, B at Z as m Z (A, B) := dim(x) i=0 ( 1) i leng OA B,Z Tor O X,Z i (O A,Z, O B,Z ). Recall the following definition of Cohen-Macaulay rings. Let R be a noetherian local ring with maximal ideal m. A sequence (x 1,, x r ) in m is called R-regular (or just regular), if the map R/(x 1,, x i 1 ) x i R/(x 1,, x i 1 ) is injective. The maximal length of R-regular sequences in m is called the depth of R, denoted by depth(r). We say R is Cohen-Macaulay if we have depth(r) = dim(r). A noetherian local ring A is called a complete intersection ring, if  can be written as  = R/(x 1,, x r ), where R is a local regular ring and (x 1,, x r ) is a R-regular sequence. A local complete intersection ring is always Cohen-Macaulay. Lemma 2.2.2. In the situation above, if O A,Z and O B,Z are Cohen-Macaulay, then Tor O X,Z i (O A,Z, O B,Z ) = 0 for i > 0, and hence m Z (A, B) = leng(o A B,Z ). Proof. Since ÔX,Z is faithfully flat over O X,Z, we have Tor O X,Z i (M, N) OX,Z Ô X,Z = TorÔX,Z i ( ˆM, ˆN), for any finitely generated O X,Z -module M, N. Since we are in the equal characteristic case, the complete regular local ring ÔX,Z = κ(z)[[x 1,, x n ]] where κ(z) is the residue field of O X,Z. We reduce to showing the following statement: Let R = k[[x 1,, x n ]], and R/I, R/J be two quotients of R of dimension n r and r respectively. If R/I and R/J are both Cohen-Macaulay and dim(r/(i + J)) = 0, then Tor R i (R/I, R/J) = 0, i > 0. Suppose first that R/I is a local complete intersection, i.e. I is generated by a regular sequence (x 1,, x r ). Let Kos (x) be the Koszul complex associated to the sequence (x 1,, x r ). Then Kos (x) is a free resolution of R/I. Hence, we have Tor R i (R/I, R/J) = H i (Kos (x) R R/J)

26 CHAPTER 2. INTERSECTION PRODUCT ON NON-SINGULAR VARIETIES The condition that dim(r/(i + J)) = dim(r/j) r implies that the image of (x 1,, x r ) in R/J is also a regular sequence. Therefore, we have H i (Kos (x) R R/J) = 0. The general case can be reduced to the previous case by diagonal trick. We consider R as a quotient of R k R = k[[1 x i, x i 1 : 1 i n]] modulo the ideal (1 x i x i 1, 1 i n). Then one has Tor R i (R/I, R/J) = Tor R k R i (R, R/I k R/J) Note that R is a quotient of R R by an ideal generated by a regular sequence, and R/I k R/J Cohen-Macaulay. Hence, we are reduced to the previous case. Theorem 2.2.3 (Serre). Let X be a smooth quasi-projective variety, and A, B X be two dimensionally transversal subschemes. Then we have [A][B] = Z m Z (A, B)[Z]. When A and B are both Cohen-Macaulay at the generic point of each irreducible component of Z, then one has [A][B] = [A B]. Remark 2.2.4. Without the Cohen-Macaulay assumption, the equality [A][B] = [A B] is false in general. Example 2.2.5 (Bézout). Let X = P n, and H 1,, H n be hypersurfaces of degrees d 1,, d n respectively. Suppose that H 1 H n contain only finitely many points, say P 1,, P m. In this case, H 1 H i is a complete intersection for each i with 1 i n, hence Cohen-Macaulay. Then we have m leng(o H1 H n,p i ) = d 1 d n. i=1 Example 2.2.6. Let X = P 4, and V 1, V 2 be two general planes meeting at a unique point p and A = V 1 V 2. Let B 1 be a plane in P 4 not passing through p. Then B 1 intersects with each of V 1, V 2 at a unique reduced point. Hence, B 1 intersects with A generically transversally, and we have deg([a][b 1 ]) = deg([a B 1 ]) = 2. Consider another plane B 2 which passes p and does not meet A elsewhere. Since B 1 is rationally equivalent to B 2, we have m p (A, B 2 ) = 2. However, one will see below that leng(o A B2,p) = 3, and hence [A][B 2 ] [A B 2 ]. The problem comes from the fact that O A,p is not Cohen-Macaulay.

2.3. PULLBACKS FOR GENERICALLY SEPARABLE FORMULAS 27 To compute leng(o A B2,p), we choose local coordinates so that ÔX,p = k[[x 1, x 2, x 3, x 4 ]], and Hence, we get I(V 1 ) = (x 1, x 2 ), I(V 2 ) = (x 3, x 4 ), I(A) = (x 1 x 3, x 1 x 4, x 2 x 3, x 2 x 4 ), I(B 2 ) = (x 1 x 3, x 2 x 4 ). O A B2,p = k[[x 1,, x 4 ]]/I(A) + I(B 2 ) = k[x 1, x 2 ]/(x 2 1, x 2 2, x 1 x 2 ), which clearly has length 3. Tor O X,p 1 (O A,p, O B2,p). The difference 3 2 = 1 comes from the length of 2.3. Pullbacks for generically separable formulas We have defined the pullback map f : A(X) A(Y ) for a flat morphism f : Y X. Using the same ideas as Moving Lemma, one can extend the definition of f to more general morphisms of quasi-projective varieties. Definition 2.3.1. Let f : Y X be a projective map of quasi-projective varieties. A subvariety V X of codimension c is called generically transverse to f if 1. f 1 (V ) is generically reduced of codimension c. 2. Y is smooth at a general point q of each irreducible component of f 1 (V ) and X and V are smooth at f(q). Note that when f : Y X is a closed immersion, then a subvariety V X generically transverse to f is equivalent to saying that V, Y are generically transverse in the sense of Definition 2.1.1. A cycle A = i m i[a i ] on X is said to be generically transverse to f if every component A i is. For such a cycle, we define f (A) = i m i [f 1 (A i )]. Recall that f : Y X is generically separable if the field extension k(y )/k(x) is separable, that is, k(y ) is a finite separable extension of purely transcendental extension of k(x). Theorem 2.3.2 (Moving Lemma for morphisms). Let f : X Y be a generically separable morphism of smooth quasi-projective varieties. (a) For any Chow class α A(Y ), there exists a cycle A Z(X) such that [A] = α and A is generically transverse to f. (b) If A Z(Y ) is a cycle generically transverse to f, the class f 1 (A) depends only on [A] A(Y ). Using Theorem 2.3.2, one can define the pullback f : A(X) A(Y ) for more general morphisms as follows.

28 CHAPTER 2. INTERSECTION PRODUCT ON NON-SINGULAR VARIETIES Theorem 2.3.3. Let f : Y X be a generically separable map of smooth projective varieties. (a) There exists a unique ring homomorphism f : A(X) A(Y ) such that if A is a subvariety of X generically transverse to f then f [A] = [f 1 (A)]. (b) (Projection formula) Let f : A(Y ) A(X) be the push-forward map of Chow groups. Then we have, for any α A(X) and β A(Y ), f (f (α)β) = αf (β). Proof. We just prove (b), since part (a) follows straightly from Theorem 2.3.2. The formula being linear in β, we may assume that β = [B] for some irreducible some variety B Y. By Moving Lemma, we may assume that B is generically transverse to the generic fibre of f, and hence generically separable to over its image f(b). We may assume α = [A], where A X is generically transverse to f B and to f(b). Note that this implies that f 1 (A) is generically transverse to B. Suppose that B is generically finite of degree d over f(b). Since A is transverse to f B, the cycle f 1 (A) B is also generically finite of the same degree d. In this case, Theorem follows as f (f (α)β) = f ([f 1 (A)]β) (A is generically transverse to f) = f ([f 1 (A) B]) f 1 (A) is generically transverse to B = d[f(f 1 (A) B)] definition of f = d[a f(b)] set theoretic equality = [A f (B)] definition of f = αf (β). The projection formula is extremely useful when computing the degree of 0-cycles. Actually, in the setup of Theorem 2.3.3, if f (α)β and αf (β) are 0-cycles, then one have (2.3.3.1) f (α)β = αf (β). Y Example 2.3.4. Consider the Segre embedding σ r,s : P r P s P (r+1)(s+1) 1 defined by ([x 0,, x r ], [y 0,, y s ]) [, x i y j, ]. Denote by Σ r,s the image of σ r,s, and we want to find deg(σ r,s ). Recall that A(P r P s ) = Z[α, β]/(α r+1, β s+1 ) X

2.3. PULLBACKS FOR GENERICALLY SEPARABLE FORMULAS 29 and A(P (r+1)(s+1) 1 ) = Z[ξ]/ξ (r+1)(s+1). By the geometric interpretation of degree, we have deg(σ r,s ) = [σ r,s, (P r P s )] ξ r+s. P (r+1)(s+1) 1 Note that σr,s(ξ) = α + β. By (2.3.3.1), we get ( ) ( ) r + s r + s deg(σ r,s ) = σr,s(ξ) r+s = α r β s =. P r P s P r P s r r Example 2.3.5. Recall that if X is a scheme, and Y is a closed subscheme of X with ideal sheaf I, the blow-up of X along Y is defined as Bl Y (X) := Proj ( I n) We have a natural projection π : Bl Y (X) Y which induces an isomorphism π 1 (X Y ) X Y. Let B denote the blow-up of P n at a point p = (1, 0,..., 0). Then B is the closed subscheme P n P n 1 defined as We have thus natural projections: n 0 B = {(x 0,..., x n ), (y 1,..., y n ) x i y j = x j y i, i, j 1}. π B α P n P n 1, where π is an isomorphism over P n p and E = π 1 (p) = P n 1. The projection α is a P 1 -fibration. It induces an isomorphism on E and its restriction on π 1 (P n p) = P n p is the projection from p: α: (x 0,..., x n ) (x 1,..., x n ). To compute the Chow ring of B, we first construct a cellular decomposition of B. Let Γ k 1 Pn 1 be the (k 1)-dimensional linear subspace given by y 1 = = y n k = 0. Then we get a flag: Γ 0 Γ 1 Γ n 2 Γ n 1 = P n 1. Put Γ k = α 1 (Γ k 1 ) B for k = 1,..., n. Let x 0 = 0. Put Λ = π 1 (Λ ), and Λ k = Γ k+1 Λ for k = 0,, n 1. Λ P n be the hyperplane given by Geometrically, Γ k is the strict transform of a k-dimensional linear subspace of P n containing p. Hence, Λ k is the preimage of a k-dimensional linear subspace of P n which does not contain p. We note that {Λ k, Γ k ; 0 k n} form a cellular decomposition of B. Indeed, Λ k Λ k 1 is clearly isomorphic to A k, and the open stratum Γ k := Γ k Γ k 1 Λ k 1 is explicitly given by Γ k = {(1, 0,, 0, λ, λy n k+2,, λy n ), (0,..., 0, 1, y n k+2,..., y n ) P n P n 1 }

30 CHAPTER 2. INTERSECTION PRODUCT ON NON-SINGULAR VARIETIES where the free variables λ, y n k+2,..., y n gives an isomorphism Γ k = A k. Hence, it follows that A(B) is generated over Z by the classes λ k = [Λ k ] for k = 0,..., n 1, and γ l = [Γ l ] for l = 1,..., n. Consider the homomorphism α : A(P n 1 ) = Z[η]/η n A(B) induced by α : B P n 1, where η is the class of Γ n 1. Then α (η l ) = γ n l for any 0 l n 1. Similarly, consider the homomorphism π : A(P n ) = Z[ξ]/ξ n+1 A(B). We have π (ξ k ) = λ n k for 0 k n. Note that, ξ is also the class of a hyperplane H P n containing p, and H is generically transverse for π. By Theorem 2.3.3, we have π (ξ) = [π 1 (H)] = [E] + [H ]. Hence, if e A 1 (B) denotes the class of the exceptional divisor E, then π (ξ) = γ n 1 + e = λ n 1 Note also that λ n 1 e = 0 because Λ n 1 E =. If we put λ = λ n 1 = π ξ, then α (η) = λ e and thus all the classes γ k and λ l can be expressed in terms of polynomials of λ and e, i.e. there exists a natural surjective map of rings θ : Z[λ, e]/(λe, λ n + ( 1) n e n ) A(B), and the degree function B : A(G) A 0(G) Z sends λ n to 1. We claim that θ is an isomorphism. Indeed, since both the source of θ is a free abelian group of rank 2n, it suffices to show that A(B) is also a free abelian group of rank 2n. It is enough to show that, for any integer k with 1 k n 1, λ k and γ k are linearly independent in A k (B). Consider the degree pairing: A k (B) A n k (B) Z : (x, y) xy Then we have { { B γ kγ n k = 0 B λ kγ n k = 1 λ B γ k λ n k = 1 B λ k(λ n k γ n k ) = 0 From this, we see easily that γ k and λ k are linearly independent. B

CHAPTER 3 LINE BUNDLES AND INTERSECTION THEORY ON SURFACES 3.1. Cartier divisors Let X be an n-dimensional variety. Let O X be the sheaf of invertible functions on X, and K(X) be the sheaf of non-zero rational functions. Recall that elements of Γ(X, K(X) /O X ) called Cartier divisors on X. Explicitly, a Cartier divisor on X is given by D = (U α, f α ) where U α form an open covering of X, and f α k(u α ) such that f α /f β in a unit on U α U β. A Cartier divisor (U α, f α ) is principal if and only if there exists a global rational function f k(x) such that f α /(f Uα ) is a unit on U α. Cartier divisors form a natural abelian group, and we denote it by Div(X). Principal Cartier divisors form a subgroup of Div(X), which we denote by Prin(X). We put Using the exact sequence one gets Pic(X) := Div(X)/ Prin(X). 1 O X K(X) K(X) /O X 1, 0 Pic(X) H 1 (X, O X ) H1 (X, K(X) ). However, it is easy to see that H 1 (X, K(X) ) = 0, and we obtain thus an isomorphism Pic(X) = H 1 (X, O X ). Recall that H 1 (X, O X ) is also the group of line bundles on X modulo rational equivalence. If D = (U α, f α ) is a Cartier divisor, then its associated line bundle O X (D) is associated to the Cech cocycle (U α, f αβ ) with f α,β = f α Uα U β f β Uα U β Γ(U α U β, O X ). Let D = (U α, f α ) be a Cartier divisor on X, and V X be a codimension 1 closed subvariety. Then we put ord V (D) = ord V (f α ) Z

32 CHAPTER 3. LINE BUNDLES AND INTERSECTION THEORY ON SURFACES for any α such that V U α. It is easy to see that this definition does not depend on the choice of α, and that ord V (D) is linear in D. Thus we get a homomorphism of groups Div(X) Z n 1 (X) given by D [D] = V X ord X(V )[V ]. Note that if D is a principal Cartier divisor associated to the rational function r k(x), then one has [D] = div(r). Hence, we get a canonical map Pic(X) A n 1 (X) = A 1 (X). Proposition 3.1.1 (EGA IV 21.6). (1) If X is normal, then the canonical maps Div(X) Z n 1 (X) and Pic(X) A 1 (X) are both injective. (2) If X is non-singular, then div(x) Z n 1 (X) and Pic(X) A 1 (X) are both isomorphisms. Proof. (1) We prove first that Div(X) A n 1 (X) is injective. Let D = (U α, f α ) be a Cartier divisor such that [D] = 0. Then by definition, this means that ord V (f α ) = 0 for any α and any closed subvariety V X such that V U α. We have to show that f α is invertible on U α for any α. We may assume that U α is affine with coordinate ring A. The condition is equivalent saying that f α A p for any prime ideal p A of height 1. Since X is normal, hence so is A. We have A = ht(p)=1 A p. It follows that f α A p = A. ht(p)=1 The injectivity of Pic(X) A n 1 (X) follows from the fact that Div(X) Rat(X) n 1 = Prin(X). (2) This is related to the fact that, if X is non-singular, every local ring of X is a unique factorial domain. Hence, every codimension 1 closed subvariety can be defined locally by 1 equation. Example 3.1.2. Let X be the planar projective curve with homogeneous equation Y 2 Z = X 3. Then one has A 0 (X) = Z, and Pic(X) A 0 (X) is surjective with kernel (k, +). Indeed, let π : X X be the normalization of X. Then one has X = P 1, and π is an isomorphism outside the singular point p = (0, 0, 1) X. Given a pint Q X, it is easy to construct an explicit rational function r on X such that div(r) = [Q] [ ], where = (0, 1, 0) X. Hence, we see that A 0 (X) = Z with a generor [ ]. To compute Pic(X), we look at the exact sequence From this, we deduce that 0 O X π O X i p, k 0. 0 k H 1 (X, O X ) H1 ( X, O X) 0. We conclude by the fact that A 0 (X) = A 0 ( X) = H 1 ( X, O X) = Z with generator O X( ).