Construction of Moduli Spaces of Stable Sheaves Via Simpson's Idea

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Abstract
The main body of the construction of the moduli space of semi-stable sheaves is how to make a quotient of a good part of a Quot-scheme by a semi-simple algebraic group. The construction in the framework of Geometric Invariant Theory, therefore, reduces to the computation of semi-stability of the good points in the Quot-scheme and the study of their orbits structure. To carry these out we had employed Gieseker’s idea, that is, a morphism of the Quot-scheme to a Gieseker’s space. Here we had used a finite resolution of a coherent sheaf by locally free sheaves and hence the underlying variety must be non-singular (see [3], [4]). To avoid this problem we can use a morphism to a product of Grassmann varieties and get moduli spaces on projective varieties which may have singularities (cf. [2]). We come then to another difficulty. In fact, if we follow this idea, then we cannot prove directly the projectivity of the moduli spaces even on a non-singular surface.

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