Clifford Theorem For Surfaces And Cayley-Bacharach Property Of Zero-dimensional Determinantal Subschemes

The classical Brill-Noether theory is to study special divisors or linear system on an algebraic curve, and the Clifford theorem is the first step of the theory (cf. [3]). The main purpose of this paper is to generalize the Clifford theorem to algebraic surfaces.In the surface case, a fundamental problem is to study the adjoint linear system |Ks+L|. Roughly speaking, the behavior of this linear system depends on the positivity of L. When L is positive, we have a celebrated method of Reider [81]. When L is zero, the canonical system has also been studied systematically by Beauville [11]. When L is negative, the linear system corresponds to the special divisors in a curve. However, in the surfaces, we have no general method to study such special divisors. In order to find a powerful method to study special linear systems in the surface case, we need to establish first the Clifford theorem. In this paper we give two generalizations of Clifford theorem from different points of view. We use these two-generalizations to define two indices a andβon a surface, like the Clifford index in the case of a curve. We study some basic properties of a andβand give some bounds for them. Then we give a detail description of the surface, when a andβare small. As an application we use our inequalities to give some bounds for the number of moduli of surfaces and generalize our techniques to get some numerical inequalities of high dimensional varieties.We also study Cayley-Bacharach property of an algebraic variety. The study of Cayley-Bacharach property of an algebraic variety has a long history in classical geometry (cf. [29]). In [88], Tan proved that the Cayley-Bacharach property of a zero dimensional complete intersection in an algebraic variety is equivalent to theκ-very ampleness of some adjoint linear systems. In [91], Tan and Viehweg generalize this result and proved that the Cayley-Bacharach property of a zero dimensional subscheme defined by the zero set of a global section of a vector bundle on an algebraic variety is equivalent to theκ-very ampleness of an adjoint linear system. The aim of this paper is to show that the result still true for the zero dimensional subscheme defined by the zero set of the wedge of some global sections of a high rank vector bundle. It generalizes a theorem of Griffiths and Harris (cf. [35], p.677). As applications, we give a construction of reflexive sheaves and vector bundles with high rank from codimension 2 subschemes, generalizing the Hartshorne-Serre correspondence. And we give lower bounds for the minimum distances of the codes associated to the zero schemes of the wedge of some global sections of vector bundles on a smooth projective variety.