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# 3 Divisors on varieties of higher dimension

For the special case of divisors (i.e., (X)) much of the picture is unchanged from that for curves. To begin with, as we saw above we have an isomorphism (X) (X).

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

In particular, deg(ch(L) . td (X))n depends only on the class cl (L) of L in (X). The Grothendieck-Hirzebruch-Riemann-Roch theorem then actually gives a method for computing (X, L) in terms of the class cl (L) in (X). However, the exact formula dimk(X,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 dimk(X,X(mD)) for large m, since (X,X(mD)) = 0 for i > 0 (by Serre's vanishing theorem), and so

dimk(X,X(mD)) = (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 3.

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

Next: 4 Zero cycles on Up: Algebraic Cycles Previous: 2 The Grothendieck-Riemann-Roch theorem
Kapil Hari Paranjape 2002-11-21