Exotic Fine Moduli Spaces of Coherent Sheaves

Abstract

Let X be a smooth projective irreducible algebraic variety over ℂ. Let S be a nonempty set of isomorphism classes of coherent sheaves on X. A fine moduli space for S is an integral algebraic structure M on the set S (i.e., S is identified with the set of closed points of M), such that there exists a universal sheaf, at least locally: there is an open cover (U i ) of M and a coherent sheaf \( \mathcal{F}_i \) on each U i × X, flat on U i , such that for every s ∈ U i , the fiber \( \mathcal{F}_{is} \) is the sheaf corresponding to s, and \( \mathcal{F}_i \) is a complete deformation of \( \mathcal{F}_{is} \).