Intersection Products

Given a regular imbedding i: X → Y of codimension d, a k-dimensional variety V, and a morphism f: V→Y, an intersection product X·V is constructed in A k-d (W), W=f -1 (X). Although the case of primary interest is when f is a closed imbedding, so W = X∩V, there is significant benefit in allowing general morphisms f. Let g: W → X be the induced morphism. The normal cone CW Vto W in V is a closed subcone of g* N X Y, of pure dimension k. We define X·V to be the result of intersecting the k-cycle [C W V] by the zero-section of g*N XY: $$ X \cdot V = {s^ * }\left[ {{C_W}V} \right] $$ where s: W → g * N X Y is the zero-section, and s* is the Gysin map constructed in Chapter 3. Alternatively X·V is the (k -d)-dimensional component of the class $$ c\left( {{g^ * }{N_X}Y} \right) \cap s\left( {W,V} \right) $$ where s (W, V) is the Segre class of W in V. If the k-cycle [C w V] is written out as a sum Σm i [C i ],with C i irreducible, one has a corresponding decomposition X·V =Σm i α i , with αi a well-defined cycle-class on the support of Ci. If the imbedding of W in V is regular of codimension d’, then E = g * N X Y/N w V is the quotient bundle, there is an excess intersection formula $$ X\cdot V = {{c}_{{d - d}}}\left( E \right) \cap \left[ W \right] $$ Given i: X → Y as above, and a morphism f: Y’→Y, form the fibre square $$ \begin{array}{*{20}{c}} {X'\mathop{{ \to Y'}}\limits^{j} } \hfill \\ {g \downarrow {{ \downarrow }^{f}}} \hfill \\ {X\mathop{{ \to Y}}\limits_{i} } \hfill \\ \end{array} $$ There are refined Gysin homomorphisms $${{i}^{!}}:{{A}_{k}}Y\prime \to {{A}_{{k - d}}}X\prime$$ determined by the formula i![V] = X·V for subvarieties Vof Y’. In this chapter the fundamental properties of these intersection operations are proved. After proving that i! is well-defined on rational equivalence classes, the most important of these properties are: (i) Compatibility with flat pull-back (§ 6.2) (ii) (i)Compatibility with proper push-forward (§ 6.2) (iii) Commutativity (§ 6.4) (iv) Functoriality (§ 6.5). For example, to calculate X • V, by (i) it suffices to calculate X • V' for any V' mapping properly and birationally to V; one may blow up V along V ∩f W to reduce to a case where the excess intersection formula applies. A particular case of (ii) is the assertion that the intersection products restrict to open subschemes: one may often compute intersection products locally. An important case of commutativity asserts that intersections may be carried out before or after specialization in a family; this will include a strong version of the “principle of continuity” in Chapter 10. When Y' = Y, i ! determines the (ordinary) Gysin homomorphisms $$ {i^*}:{A_k}Y \to {A_{k - d}}X$$ Functoriality (iv) refines the statement that (j i)* = i* j* for i: X → Y, j: Y→Z regular embeddings. More generally, if f: X→Y is a local complete intersection morphism, there are Gysin homomorphisms f*, and refined homomorphisms f ! .These Gysin homomorphisms are used to describe the group A * $$ \widetilde Y$$ ,when $$ \widetilde Y$$ is the blow-up of a scheme Y along a regularly imbedded subscheme. A new blowup formula describes the Gysin map from A * Y to A * $$ \widetilde Y$$ explicitly. The rest of this book is based on this intersection product and the fundamental properties proved in § 6.1— § 6.5. As in Chap. 2, the formal properties can be motivated from topology. As we shall see in Chap. 19, a regular imbedding X SY of codimension d determines an orientation, or generalized Thom class, in H 2d (Y, Y—X). The Gysin maps are the algebraic geometry versions of cap product by this orientation class, or with its pull-back to Y', if Y' maps to Y.