Divisor class groups of singular surfaces

Abstract

We compute divisors class groups of singular surfaces. Most notably we produce an exact sequence that relates the Cartier divisors and almost Cartier divisors of a surface to the those of its normalization. This generalizes Hartshorne's theor em for the cubic ruled surface in P 3 . We apply these results to limit the possible curves that can be s et-theoretic complete intersection in P 3 in characteristic zero. On a nonsingular variety, the study of divisors and linear systems is classical. In fact the entire theory of curves and surfaces is dependent on this study of codimension one subvarieties and the linear and algebraic families in which they move. This theory has been generalized in two directions: the Weil divisors on a normal variety, taking codimension one subvarieties as prime divisors; and the Cartier divisors on an arbitrary scheme, based on locally principal codimension one subschemes. Most of the literature both in algebraic geometry and commutative algebra up to now has been limited to these kinds of divisors. More recently there have been good reasons to consider divisors on non-normal varieties. Jaffe (9) introduced the notion of an almost Cartier divisor, which is locally principal off a subset of codimension two. A theory of generalized divisors was proposed on curves in (14), and extended to any dimension in (15). The latter paper gave a complete description of the generalized divisors on the ruled cubic surface in P 3 . In this paper we extend that analysis to an arbitrary integra l surface X, explaining the group APicX of linear equivalence classes of almost Cartier divisors on X in terms of the Picard group of the normalization S of X and certain local data at the singular points of X. We apply these results to give limitations on the possible curves that can b e set-theoretic compete intersections in P 3 in characteristic zero In section 2 we explain our basic set-up, comparing divisors on a variety X to its normalization S. In Section 3 we prove a local isomorphism that computes the group of almost Cartier divisors at a singular point of X in terms of the Cartier divisors along the curve of singulari ties and its inverse image in the normalization. In Section 4 we derive some global exact sequences for the groups PicX, APicX, and PicS, which generalize the results of (15, §6) to arbitrary surfaces These results are particularly transparent for surfaces wi th ordinary singularities, meaning a double curve with a finite number of pinch points and triple points.