NOTES ON INTERSECTION THEORY 1. Motivation ...

NOTES ON INTERSECTION THEORY HEINRICH HARTMANN

1. Motivation from Topology Intersection theory addresses the following question: Given a space X and a collection of subspaces Z1 , . . . , Zn ⊂ X. How many points lie in the intersection Z1 ∩ · · · ∩ Zn ? In general this question very hard to answer. More can be said if we modify our question a bit: (1) Assume X is a smooth and compact manifold. (2) Assume Zi admit triangulations and meet transversally. (3) Choose orientations on X and Zi and count the number of points in Z1 ∩ · · · ∩ Zn with the appropriate signs. Under this assumptions the signed count of intersection points does not change if we move Zi in their homology class. It turns out that every collection of (singular) sub-manifolds Zi can be made transversal inside their homology classes and that we can define a product structure on the homology groups by “taking transversal intersections”: [Z1 ] · [Z2 ] = [Z10 ∩ Z20 ] where Zi0 are representatives of [Zi ] meeting transversally. In this way we reformulated our initial question into computing the degree of a certain 0-dimensional homology cycle #or Z1 ∩ · · · ∩ Zn = deg([Z1 ] · · · · · [Z1 ]). There are many topological tools to compute these intersection products. The most important ones being (Poincare-) duality and the theory of characteristic classes in the case Zi are vanishing loci of sections of vector bundles. This picture remains basically the same if we change our setting from differentiable manifolds to complex analytic spaces. Here are the main differences. (1) Every analytic space carries a canonical orientation. So all the signs introduced above disappear. (Good news!) (2) Every analytic space admits a triangulation. (Good news!) (3) Not every intersection can be made transversal inside a family of analytic spaces. (Bad News!) One can still use topological deformations (introducing maybe negative orientations) or replace Zi by a linear combination of movable cycles (which may have negative coefficients). Date: 30.04.2010. 1

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(4) Analytic singularities are not too bad. In many cases one can assign an “intersection multiplicity” or an “excess-intersection class” to a component of a degenerate intersection which measures the local contribution to the intersection product. In the algebraic setting of schemes we cannot use the topological theory alluded to above. Instead one has to redo the whole theory using algebraic methods only. In my opinion it is nevertheless very helpful to keep the topological motivation in mind. In this note we will review the some aspects of this algebraic theory of intersections following Fulton’s book [Ful84]. References [Ful84] William Fulton, Intersection theory, second edition, Springer 1998

2. Chow Groups A “scheme” is for us a scheme of finite type over a fixed field κ in the sense of in Hartshorn’s book. A variety is a irreducible and reduced scheme. Definition 2.1. [Ful84, Section 1.3] Let X be a scheme the group of kdimensional cycles is the free abelian group generated by all k-dimensional subvarieties. Zk (X) = Zh[Z] | Z ⊂ X k-dimensional variety i Let W ⊂ X be a k + 1-dimensional variety, and f ∈ K(W ) a rational function on W . There is an associated divisor on W X div(f ) = ordZ (f )[Z] ∈ Zk (W ). Z⊂W

The sum runs over all k-dimensional sub-varieties Y of W and ordZ (f ) is the order function on K(W ) defined by the valuation ring OW,Z ⊂ K(W ). Geometrically this is the order of vanishing of f along V . The equivalence relation on Zk (X) generated by div(f ) ∼ 0 for all k + 1dimensional sub-varieties W ⊂ X and f ∈ K(W ) is called rational equivalence. The quotient Ak (X) = Zk / ∼ is called the k-dimensional Chow group of X. We set A∗ (X) = ⊕k Ak (X) and A∗ (X) = ⊕k Ak (X), Ak (X) = An−k (X) if X is of pure dimension n. On Chow groups we have the following basic operations. (1) Proper push-forward [Ful84, Section 1.4]. For a proper morphism f : X → Y there is a unique homomorphism f∗ : Ak (X) → Ak (Y ) with the property that if Z ⊂ X is a sub-variety and W = f (Z) (which is a subvariety of Y ) then f∗ [Z] = deg(Z/W )[W ]. (2) Fundamental cycles [Ful84, Section 1.5]. A subscheme Y ⊂ X determines a cycle [Y ] ∈ A∗ (X) by X [Y ] = mi [Yi ] i

where Yi are the irreducible components of Y and mi = length(OY,Yi ) are the geometric multiplicities of Yi in Y .

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(3) Flat pull-back [Ful84, Section 1.6]. For a flat morphism f : X → Y of constant relative dimension n there is a unique homomorphism f ∗ : Ak (Y ) → Ak+n (X) mapping a sub-variety Z ⊂ Y to the fundamental cycle of the inverse image scheme f −1 (Z). (4) Gysin morphisms [Ful84, Chapter 6]. Let i : X → Y be a regular embedding of codimension d then there is a Gysin morphism i∗ : Ak (Y ) → Ak−d (X). If Z ⊂ Y is a variety intersecting X properly then i∗ ([Z]) can be represented by a linear combination of the irreducible components of the intersection scheme. If Z intersects X transversally then i∗ ([Z]) = [Z ∩ X] is the fundamental cycle of the (reduced) intersection scheme. The moving lemmas given below can be used to reduce the general definition to this cases. There is also a direct construction due to Fulton which uses a deformation to the normal cone. (5) Intersection Product [Ful84, Chapter 8]. If X is a non-singular variety of dimension n then there is a product on A∗ (X). In this case the diagonal embedding δ : X → X × X is a regular embedding and the product is defined as α · β = δ ∗ (α × β) for α, β ∈ A∗ (X). Proposition 2.2 (Moving lemma). [Ful84, Section 11.4]. If X is nonsigular, quasi-projective and α and β are cycles on X then there exists a cycle β 0 ∼ β such that α and β 0 intersect properly. If moreover κ is algebraically closed there is a β 00 ∼ β such that α and β 00 meet transversally. There are various compatibilities between these operations. 3. Chern Classes Let L be a line bundle on a scheme X we define a first Chern operation c1 (L) ∩ : Ak (X) −→ Ak−1 (X) by the following procedure. For a k-dimensional subvariety Z ⊂ X the line bundle L|Z can be represented by a Cartier divisor D, with OZ (D) = L|Z . The associated Weil divisor is also denoted also by D and defines a k − 1cycle [D] on X. This cycle is unique up to rational equivalence, and thus we can define c1 (L) ∩ [Z] = [D]. To construct higher Chern operators we consider projectivized bundles. Let E be a vector bundle over X of rank r > 0, we denote by π : P(E) = P roj(Sym∗OX E ∨ ) −→ X the associated bundle of projective spaces. It comes with a canonical line bundle OP(E) (1) which is relatively ample and restricts fiber-wise to the canonical bundle on the projective space P(Ex ). Proposition 3.1. Let ξ = c1 (OP(E) (1)), then the morphism π ∗ : A∗ (X)h1, ξ, . . . , ξ r−1 i −→ A∗ (P(E)),

α ⊗ ξ k 7→ ξ k ∩ π ∗ α

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is an isomorphism of A∗ (X) modules. We define higher Chern operations in the following way. For α ∈ Ak (X) we consider π ∗ (ξ r ∩ α) ∈ A∗ (P(E)) ∼ = A∗ (X)h1, ξ, . . . , ξ r−1 i which we decompose according to the proposition in classes −ξ r ∩ π ∗ α = ξ 0 ∩ π ∗ αr + ξ 1 ∩ π ∗ αr−1 + · · · + ξ r−1 ∩ π ∗ α1 and set ci (E) ∩ α = αi . Thus we get the fundamental relation r X

(1)

ξ r−p π ∗ (cp (E) ∩ α) = 0

p=0

with c0 (E) ∩ α := α in A∗ (P(E)). We also define Chern cycle classes as cp (E) = cp (E) ∩ [X]. If X is a nonsingular variety the Chern operations can be reconstructed from the Chern classes via ci (E) ∩ α = ci (E) · α. Chern classes have the following geometric significance. Proposition 3.2. [Ful84, 14.4.2] Let X be a variety and assume there are s1 , . . . , sr−p+1 general sections1 of E then cp (E) = [{ x ∈ X | s1 (x), . . . , sr−p+1 (x) linearly dependent }]. In particular c1 (E) = [{ x ∈ X | det(s1 (x), . . . , sr (x)) = 0 }] = c1 (det(E)), cr (E) = [{ x ∈ X | s1 (x) = 0 }]. Warning: It is not true that cp (E) = ctop (Λr−p+1 (E)) since s1 ∧ · · · ∧ sr−p+1 is not a generic section. If X is not a variety and si are not general there are localized Chern classes which are supported on the corresponding zero-schemes and push forward to the Chern classes constructed in X. Remark 3.3. The vector bundle π ∗ (E) ⊗ O(1) has a canonical nowherevanishing section. It follows from this that cr (π ∗ (E) ⊗ O(1)) = 0. using the formula for chern classes of tensor products with line bundles, we derive at the fundamental relation (1). 4. Cones and Segre Classes Definition 4.1. Let E be a vector bundle on a scheme X of dimension n. The Segre operations s0 (E), . . . , sn (E) are defined to be inverse to the Chern operations st (E) = s0 (E)t0 + · · · + sn (E)tn := 1/ct (E). In particular s0 (E) = [X], s1 (E) = −c1 (E), s2 = −c2 (E) + c1 (E)2 . The Segre classes are defined as sp (E) = sp (E) ∩ [X] ∈ A∗ (X). 1This means for us the intersection schemes defining c have the expected dimension p

and a suitable depth-condition is satisfied. If X is Cohen-Macaulay (e.g. smooth), κ is algebraically closed and E is globally generated then there are always general sections [Ful84, B.9].

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Why Segre classes? (1) The Segre classes (not the operations) admit a generalization to cones, which is used e.g. in Fulton’s definition of the intersection product (and Daniel’s talk today). (2) The Chern classes are more convenient for computations since cp (E) = 0 for p > rk(E). (3) Segre classes also have a geometric interpretation similar to Proposition 3.2. Proposition 4.2. [Ful84, 14.3.2, 14.4.2] Let X be a variety and assume there are s1 , . . . , sr+p−1 general sections of E then (−1)p sp (E) = [{ x ∈ X | rk(s1 (x), . . . , sr+p−1 (x)) ≤ r − 1 }]. Proposition 4.3. Let E be a vector bundle of rank r > 0 over X and π : P(E) → X the associated projective bundle. Denote by ξ = c1 (OP(E) (1)) the first Chern operation. Then sp (E) = π∗ (ξ r+p−1 ∩ [P(E)]). Note that π∗ (ξ k ∩ [P(E)]) = 0 for k < r − 1 = dim(P(E)/X). P Proof. Define s˜t (E) = np=0 π∗ (ξ r+p−1 ∩ [P(E)])tp , n = dim(X). We want to show s˜t (E) = st (E) or equivalently s˜t (E) ct (E) = 1. We calculate r n X X s˜t (E) ct (E) = cq (E)tq π∗ (ξ r+p−1 ∩ [P(E)])tp q=0

=

p=0

X X

π∗ (π ∗ cq (E)ξ r+p−1 ∩ [P(E)])td

q ≤ r, p ≤ n, p, q ≥ 0

d p+q=d

= π∗ (ξ r−1 ∩ [P(E)])t0 +

X X

π∗ ((π ∗ cq (E)ξ r−q )ξ d−1 ∩ [P(E)])

d>0 p+q=d 0

= [X]t + 0. In the last step we used the fundamental relation (1) and the fact that π∗ (ξ r−1 ∩ [P(E)]) = [X]. The latter is inituitively clear since fiber-wise c1 (O(1))r−1 ∩ [Pr−1 ] is represented by a point. A proof can be found in [Ful84, Prop. 3.1.].  We now introduce the notion of cones. Definition 4.4. Let X be a scheme and S ∗ , ∗ ≥ 0 a graded OX -algebra satisfying the following conditions. (1) The canonical morphism OX → S 0 is an isomorphism. (2) It is S ∗ generated by S 1 as an S 0 -algebra. We define the associated cone to be π : C = SpecX (S ∗ ) −→ X, the projective cone q : P(C) = ProjX (S ∗ ) −→ X and the projective closure of C as P(C ⊕ 1) = ProjX (S ∗ [z]), where the grading on the polynomial ring S ∗ [z] is determined by deg(z) = 1. Also z determines a section of the canonical line bundle O(1)P(C⊕1) . Its zero

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scheme is canonically isomorphic to P(C) and the complement to C. The Cartier divisor P(C) ⊂ P(C ⊕ 1) is called hyperplane at infinity. Definition 4.5. Let C be a cone over a scheme X and q : P(C ⊕1) → X the projective closure of C with canonical bundle OP(C⊕1) (1). Then we define the Segre class of C as X s(C) = q∗ ( c1 (OP(C⊕1) (1))k ∩ [P(C ⊕ 1)]). k≥0

Remark 4.6. If E = C is a vector bundle of positive rank, then s(C) = P s (E). This follows form the fact that P(C) ⊂ P(C ⊕ 1) is a Cartier p p divisor in class OP(C⊕1) (1) and Proposition 4.3. By the same argument we may replace P(C ⊕ 1) by P(C) in the definition of s(C) if C is pure dimensional and there are no components Ci of relative dimension zero over X. We always have s(C) = s(C ⊕ 1). Definition 4.7. Let Y ⊂ X be a subscheme of X defined by an ideal I ⊂ OX . We define a graded algebra ⊕k≥0 I k /I k+1 which is easily seen to fulfill the assumptions of definition 4.4. The associated cone M C(Y /X) = SpecX I k /I k+1 −→ Y k

is called normal cone of Y in X and s(Y /X) = s(C(Y /X)) is called the Segre class of Y in X. Before we comment on the geometric interpretation we establish some basic properties of Segre classes of sub-schemes. Lemma 4.8. [Ful84, Lem. 4.2] Let X be a purely n-dimensional scheme, X1 , . . . , Xr the (reduced) irreducible components of X and mi the geometric multiplicity of Xi in X. If Y ⊂ X is a subscheme and Yi = Y ∩ Xi (as schemes) then X s(Y /X) = mi s(Yi /Xi ). i

Proof. As we will need the following construction later on, we give a rather detailed proof. Let MY X be the blowup of Y × {0} in X × A1 . The exceptional divisor of MY X is EY X = P(C(Y × {0}/X × A1 )) = P(C(Y /X) ⊕ 1). Note that this scheme occures in the definition of the Segre class. We will reduce the proposition to an equality between Cartier divisors on MY X. We claim that the irreducible components of MY X are MYi Xi with multiplicities mi . Indeed, as Y × {0} in X × A1 is nowhere dense every irreducible component of X × A1 meets X × A1 \ Y × {0}. On the other hand the exceptional divisor E ⊂ MY E is Cartier and hence contains no irreducible components. Hence we can compare the components (and multiplicities) on the complement of the exceptional locus, where canonical map π : MY X → X × A1 is an isomorphism. It remains to show that the strict transforms of the irreducible components Xi ×A1 of X ×A1 are given by MYi Xi . There is a closed embedding MYi Xi →

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MY X ×X Xi by compatibility of blowup and pullback [Ful84, B.6.9]. We compose this morphism with the closed embedding MY X ×X Xi → MY X. Now MYi Xi is irreducible and maps birationally onto Xi × A1 . This shows the claim. The exceptional divisor EY X of MY X restricts to the exceptional divisors EYi Xi of MYi Xi [Ful84, B.6.9]. As MY X is pure dimensional and the exceptional divisor E is Cartier, we can apply [Ful84, Lemma 1.7] which tells us that X X [EY X] = mi [EY X ∩ MYi Xi ] = mi [EYi Xi ] i

i

as divisors on MY X. This yields the equality X [P(C(Y /X) ⊕ 1)] = mi [P(C(Yi /Xi ) ⊕ 1)] i

of fundamental classes in Zn (MY X). Capping with c1 (O(1))k and pushing down to X we get the required identity of Segre classes.  Proposition 4.9. [Ful84, Prop. 4.2] Let X and X 0 be pure dimensional schemes, f : X 0 → X a morphism, Y ⊂ X a subscheme, with inverse image scheme f −1 (Y ) = Y 0 and g : Y 0 → Y the restriction of f . (1) If f is proper X irreducible and f maps the irreducible components of X 0 onto X, then g∗ s(Y 0 /X 0 ) = deg(f )s(Y /X). (2) If f is flat2 then g ∗ s(Y /X) = s(Y 0 /X 0 ). P Proof. Ad (1). The degree of f is defined to be deg(f ) = i mi deg(fi ) where Xi0 are the irreducible components of X 0 and fi : Xi0 → X the restriction of f . (Hence fi is a map between varieties and the usual definition of degree applies). Using the Lemma we can easily reduce to the case that X 0 is irreducible. With the notation of the proof of the Lemma. Let M = MY X and 0 M = MY 0 X 0 , with exceptional divisors E = EY X and E 0 = EY 0 X 0 . We have a canonical map F : MY 0 X 0 → MY X induced by f : X 0 → X. It is F ∗ [E] = [E 0 ] as Cartier divisors on MY 0 X 0 . This follows from [Ful84, B.6.9] since Y 0 → X 0 is the pullback of Y → X along f : X 0 → X. It is F∗ [MY 0 X 0 ] = d[MY X], d = deg(f ) and by the projection formula we find F∗ [E 0 ] = F∗ (c1 (F ∗ OM (E)) ∩ [M 0 ]) = c1 (OM ([E])) ∩ F∗ [M 0 ] = d[E]. Let G : E 0 → E be the restriction of F to the exceptional divisors. Under the identifications E = P(C(Y /X) ⊕ 1) and E 0 = P(C(Y 0 /X 0 ) ⊕ 1) we have G∗ OP(C(Y /X)⊕1) (1) = OP(C(Y 0 /X 0 )⊕1) (1). This follows form the fact that G is induced by a mapping between the normal cones which is compatible with the Gm action on the fibers (i.e. the grading of the sheaves of algebras). 2We always assume flat morphisms to have a constant relative dimension.

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The proposition follows now form the following straight forward computation. Denote by q : E → X and q 0 : E 0 → X the canonical bundle projections. We have X X g∗ s(Y 0 , X 0 ) = g∗ q∗0 c1 (OE 0 (1))k ∩ [E 0 ] = q∗ G∗ c1 (G∗ OE (1))k ∩ [E 0 ] k

= q∗

X

k 0

k

c1 (OE (1)) ∩ G∗ [E ] = dq∗

X

k

c1 (OE (1))k ∩ G∗ [E]

k

= ds(Y, X). For (2) we conclude similarly X X g ∗ s(Y /X) = g ∗ q∗ c1 (OE (1))k ∩ [E] = q∗0 G∗ c1 (OE (1))k ∩ [E] k

=

q∗0

X

k

c1 (OE 0 (1)) ∩ [E ] = s(Y /X 0 ). k

0

0



k

We showed in particular birational invariance of Segre classes. Corollary 4.10. If f : X 0 → X is a birational morphism between varieties, and Y ⊂ X and arbitrary subscheme with pre-image scheme f −1 (Y ) = Y 0 , then f∗ s(Y 0 /X 0 ) = s(Y /X). 4.1. Examples. (1) If Y ⊂ X is a regular embedding, then the normal cone is a vector bundle CY X = NY X and the Segre class is given by s(Y /X) = c(NY X)−1 ∩ [Y ]. (2) A point P ∈ X is regularly embedded if and only if X is smooth at P . In this case its Segre class is s(P/X) = [P ]. If P is not a smooth point, then s(P/X) = (eP X)[P ] where ep X is Samuel’s multiplicity defined as the leading coefficient of the polynomial t 7→ length(OX,P /mtP ), t  0 divided by n!. (3) Let Y ⊂ X be a proper closed subscheme of a variety X. Let π : ˜ → X be the blowup of X along Y . The exceptional divisor E is X Cartier and hence regularly embedded with normal bundle OX˜ (E)|E . Combining this with birational invariance we find X s(Y /X) = π∗ (−c1 (O(E)))k ∩ [E] k

X ˜ = π∗ (−1)k c1 (O(E))k+1 ∩ [X]. k

This is Segre’s original definition. Example 4.11. The cusp X = {x2 − y 3 } ⊂ A2 is resolved by f : A1 → C, t 7→ (t3 , t2 ) = (x, y). Let P = {x = y = 0} then f ∗ ([P ]) = [t2 = t3 = 0] = 2[t = 0] ∈ A1 (A1 ). This shows eP X = 2.

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Let π : X 0 → X be the blowup of X = A2 in P = {x = y = 0}. Let ˜ be the exceptional divisor. The proposition says in this case that E⊂X ˜ = π∗ c(NE X) ˜ −1 ∩ [E] = π∗ (−c1 (O(E)) ∩ [E]). deg(π)s(P/X) = π∗ s(E/X) Using s(P/X) = [P ], we can deduce the birationality (deg(π) = 1) from the self intersection formula E 2 = −1 and vice versa. 5. Gauss–Bonnet formula Theorem 5.1. Let X be a proper, non-singular variety of dimension n over the complex numbers κ = C then Z cn (T X) = χtop (X) P where T X is the tangent bundle on X and χtop (X) = ni=0 hi (X an , Q) is the topological Euler characteristic of the associated complex manifold. We R denote the degree of a zero-cycle α ∈ A0 (X) by α = deg(α). We give a sketch of an algebraic proof to this topological theorem using some more advanced techniques. Proof. Let ∆ ⊂ X × X be the diagonal. We show that both sides of the equation are equal to deg([∆] · [∆]). Step 1. We claim that Z cn (T X) = deg([∆] · [∆]). Let i : X → ∆ ⊂ X × X be the inclusion, then T X = i∗ N∆ (X × X) by definition. By the excess intersection formula we find [∆] · [∆] = i∗ i∗ [∆] = i∗ (cn (N∆ (X × X)) ∩ [X]) = i∗ (cn (T X) ∩ [X]). Step 2. It is deg([∆].[∆]) = χ(O∆ ⊗L O∆ ). This can be seen as follows. Z L χ(O∆ ⊗ O∆ ) = ch(O∆ ⊗L O∆ ) · td(X × X) Z = ch(O∆ ) · ch(O∆ ) · td(X × X) Z = ([∆] + α) · ([∆] + α) · (1 + β) Z = [∆] · [∆] The first step is Hirzbruch-Riemann-Roch theorem. The second step follows form the fact that ch : K(X) → A∗ (X)Q is a ring homomorphism. The third step uses Grothendieck-Riemann-Roch for embeddings i : Y → X: ch(i∗ OY ) = i∗ (ch(OY ) ∩ td(NY X)). Recall that the Todd class of a bundle is always td(E) = 1 + β with β of higher codimension. This justifies the last step. Step 3. χ(O∆ ⊗L O∆ ) = χtop (X). The cohomology sheaves of O∆ ⊗L O∆ are the denoted by T ori (O∆ , O∆ ) = Hi (O∆ ⊗L O∆ ) it follows X χ(O∆ ⊗L O∆ ) = (−1)i χ(T ori (O∆ , O∆ )). i

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Computing the Tor sheaves locally using a Koszul resolution one finds T or (O∆ , O∆ ) ∼ = Λi (N∆ (X × X)∨ ). i

Note that Λi (N∆ (X × X)∨ ) = Λi (T X ∨ ) = Ωi X. Now the theorem follows from GAGA and the Hodge decomposition. X X X X hq (X an , Ωp,an ) (−1)n (−1)p+q hq (X, Ωp ) = (−1)p χ(Ωp ) = n

p,q

p

=

X n

n n

(−1) h (X, C) = χ

top

(X).

p+q=n