Divisors

Abstract

If D is a Cartier divisor on a scheme X, and αis a k-cycle on X, we construct an intersection class

$$ D \cdot \alpha \in {A_{k - 1}}\left( {\left| D \right| \cap \left| \alpha \right|} \right) $$

where ${\left| D \right|^1}$, $\left| \alpha \right|$ are the supports of D and α. For α = [V], V a subvariety, D · [V] is defined by one of two procedures: (i) if $V \not\subset \left| D \right|$, D restricts to a Cartier divisor on V, and D · [V] is defined to be the associated Weil divisor of this restriction; (ii) if $V \subset \left| D \right|$, the restriction of the line bundle ${\theta _x}\left( D \right)$ to V is the line bundle of a well-defined linear equivalence class of Cartier divisors on V, and D · [V] is represented by the associated Weil divisor of any such Cartier divisor.

We prove that if a is rationally equivalent to zero on X, then D·α is zero in ${A_{k - 1}}\left( {\left| D \right|} \right)$ ; there are therefore induced homomorphisms

$$ {A_k}X \to {A_{k - 1}}\left( {\left| D \right|} \right) $$

In the special but important case where D is the inverse image of a point for a morphism from X to a smooth curve, D·α is the specialization of α; in this case (or whenever D is principal) D·α can be well-defined as a cycle, setting $D \cdot \left[ V \right] = 0$ if $V \subset D$ The above fact therefore includes the assertion that rational equivalence is preserved under specialization.

If D and D’ are Cartier divisors on a scheme X, and a is α k-cycle on X,a crucial property is the commutative law

$$ D \cdot \alpha \in {{A}_{{k - 1}}}\left( {\left| D \right| \cap \left| \alpha \right|} \right)$$

in ${A_{k - 2}}\left( {\left| D \right| \cap \left| {D'} \right| \cap \left| \alpha \right|} \right)$ Consider, for example, the case where $f:X \to {\mathbb{A}^2}$ is a morphism, and D and D’ are the inverse images of the two axes. One may specialize a cycle first to the part of X over the x-axis, and then specialize the resulting cycle to f ^-1(0); or one may first specialize over the y-axis, then over the origin. The resulting cycles one arrives at by these two routes may well be different², but the above says they are rationally equivalent.

Both of the above facts follow from the identity (Theorem 2.4):

$$ D \cdot \left[ {D'} \right] = D' \cdot \left[ D \right] in {{A}_{{n - 2}}}\left( {\left| D \right| \cap \left| {D'} \right|} \right)$$

for Cartier divisors D, D’ on an n-dimensional variety X, with [D], [D’] their associated Weil divisors.

A Cartier divisor D on a scheme X determines a line bundle $L = {\copyright_X}(D)$ and a trivialization of L over X—|D| . Only the line bundle, the support, and the trivialization are needed to carry out the above intersection construction’. These concepts are formalized in the notation of a pseudo-divisor (§ 2.2); there is the added advantage that a pseudo-divisor, unlike the stricter notion of a Cartier divisor, pulls back under arbitrary morphisms

Intersecting with divisors is used to construct homomorphisms

$$ {A_k}X \to {A_{k - 1}}X,\alpha \to {c_1}(L) \cap \alpha , $$

for a line bundle L on X, and to construct Gysin homomorphisms

$$ {i^*}:{A_k}X \to {A_{k - 1}}D$$

when i is the inclusion of an effective Cartier divisor D in X. These operations will be generalized to higher codimension in subsequent chapters.

Article Footnote

¹This shorthand for Supp (D) should not be confused with a notation for complete linear systems, which do not occur in this chapter.

² This corresponds to the fact that a family of cycles parametrized by a smooth parameter variety has a unique limiting cycle over a missing point when the parameter variety is a curve, but not when it has dimension two or more.