3 Divisors on varieties of higher dimension

For the special case of divisors (i.e., $\CH^{1}_{}$(X)) much of the picture is unchanged from that for curves. To begin with, as we saw above we have an isomorphism $\Pic$(X) $\cong$ $\CH^{1}_{}$(X).

When X is projective one can define an equivalence relation on $\CH^{1}_{}$(X) as follows. Let C be any smooth curve and D $\subset$ X×C be a divisor which does not contain any fibre of X×C$\to$C. For any pair of points p,q on C the divisor D $\cap$ X×{p} - D $\cap$ X×{q} can be considered as a divisor on X, which is said to be algebraically equivalent to 0. The quotient of $\CH^{1}_{}$(X) by this equivalence relation is a finitely generated Abelian group--the Néron-Severi group $\NS$(X). This gives us a generalisation of the degree homomorphism for curves, namely the quotient map cl : $\CH^{1}_{}$(X)$\to$$\NS$(X).

Upto torsion this equivalence relation can also be defined using intersection theory. We define a divisor D to be numerically equivalent to zero if the intersection number (D ^. C) = 0 for every curve C contained in X. Then one knows that some multiple of D is in fact algebraically equivalent to 0. Conversely, if a divisor D is algebraically equivalent to 0 then it is also numerically equivalent to 0. In case the ground field is $\Bbb$C then we can also identify algebraic equivalence with homological equivalence: i.e., a divisor is algebraically equivalent to 0 precisely if it lies in the kernel of the cycle class map $\CH^{1}_{}$(X)$\to$$\HH^{2}_{}$(X,$\Bbb$Z).

In particular, deg(ch(L) ^. td (X))_n depends only on the class cl (L) of L in $\NS$(X). The Grothendieck-Hirzebruch-Riemann-Roch theorem then actually gives a method for computing $\chi$(X, L) in terms of the class cl (L) in $\NS$(X). However, the exact formula dim_k$\HH^{0}_{}$(X,$\cal {O}$_X(D)) = deg D + 1 - g, valid for divisors of large degree on a curve of genus g, has only a partial generalisation to higher dimensions: if D is an ample divisor, then the Grothendieck-Riemann-Roch theorem gives a formula for dim_k$\HH^{0}_{}$(X,$\cal {O}$_X(mD)) for large m, since $\HH^{i}_{}$(X,$\cal {O}$_X(mD)) = 0 for i > 0 (by Serre's vanishing theorem), and so

dim_k$$\HH^{0}_{}$$(X,$$\cal {O}$$_X(mD)) = $$\chi$$($$\cal {O}$$_X(mD)) = deg(ch(L)td (X))_n  .

For effective divisors D on a surface, the general case was studied by Zariski[43], and its solution is completed in [11], where a similar problem is posed for suitable divisors on varieties of dimension $\geq$ 3.

The collection of all effective divisors on X corresponding to a fixed class c in $\NS$(X) form a projective scheme Hilb_c(X). Also, in analogy with the case for curves, the kernel A¹(X) of the morphism $\CH^{1}_{}$(X)$\to$$\NS$(X) is also naturally isomorphic to (the group of k-rational points of) an Abelian variety, the Picard variety $\Pic^{0}_{}$(X). Fixing one divisor C in the class c we obtain a natural morphism Hilb_c(X)$\to$$\Pic^{0}_{}$(X), the Abel-Jacobi morphism. The fibres of this morphism precisely consist of effective divisors corresponding to a fixed class in $\CH^{1}_{}$(X). As in the case of curves, one can show that for a ``sufficiently large'' multiple of an ample class c the morphism Hilb_{m ^. c}$\to$$\Pic^{0}_{}$(X) is surjective.