**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
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 *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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