Let X be a projective irreducible smooth algebraic
variety. A
fine moduli space of sheaves on X is a family
of coherent sheaves on X parametrized by an integral variety M
such that :
is flat on M; for all distinct points x, y of M
the sheaves
,
on X are not isomorphic and
is a complete deformation of
;
has an obvious local universal property. We define also a fine moduli
space defined locally, where
is replaced by a family
,
where
is defined on an open subset Ui of M, the Ui
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.