Varietes de modules alternatives

Abstract

Let X be a projective irreducible smooth algebraic variety. A "fine moduli space" of sheaves on X is a family F of coherent sheaves on X parametrized by an integral variety M such that : F is flat on M; for all distinct points x, y of M the sheaves F_x, F_y on X are not isomorphic and F is a complete deformation of F_x; F has an obvious local universal property. We define also a "fine moduli space defined locally", where F is replaced by a family (F_i), where F_i is defined on an open subset U_i of M, the U_i covering M. This paper is devoted to the study of such fine moduli spaces. We first give some general results, and apply them in three cases on the projective plane : the fine moduli spaces of prioritary sheaves, the fine moduli spaces consisting of simple rank 1 sheaves, and those which come from moduli spaces of morphisms. In the first case we give an exemple of a fine moduli space defined locally but not globally, in the second an exemple of a maximal non projective fine moduli space, and in the third we find a projective fine moduli space consisting of simple torsion free sheaves, containing stable sheaves, but which is different from the corresponding moduli space of stable sheaves.