Variétés de modules alternatives. Annales de l'Institut Fourier (1999) , 57-139

Let X be a projective irreducible smooth algebraic variety. A fine moduli space of sheaves on X is a family ${\mathcal F}$ of coherent sheaves on X parametrized by an integral variety M such that : ${\mathcal F}$ is flat on M; for all distinct points x, y of M the sheaves ${\mathcal F}_x$ , ${\mathcal F}_y$ on X are not isomorphic and ${\mathcal F}$ is a complete deformation of ${\mathcal F}_x$ ; ${\mathcal F}$ has an obvious local universal property. We define also a fine moduli space defined locally, where ${\mathcal F}$ is replaced by a family $({\mathcal F}_i)$ , where ${\mathcal F}_i$ 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 example of a fine moduli space defined locally but not globally, in the second an example 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.

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