Moduli of vector bundles on primitive multiple schemes

Abstract

A primitive multiple scheme is a Cohen–Macaulay scheme Y such that the associated reduced scheme X = Yred is smooth, irreducible, and that Y can be locally embedded in a smooth variety of dimension [Formula: see text]. If n is the multiplicity of Y, there is a canonical filtration [Formula: see text], such that [Formula: see text] is a primitive multiple scheme of multiplicity i. The simplest example is the trivial primitive multiple scheme of multiplicity n associated to a line bundle L on X: it is the nth infinitesimal neighborhood of X, embedded in the line bundle [Formula: see text] by the zero section. The main subject of this paper is the construction and properties of fine moduli spaces of vector bundles on primitive multiple schemes. Suppose that [Formula: see text] is of multiplicity n, and can be extended to [Formula: see text] of multiplicity [Formula: see text], and let [Formula: see text] be a fine moduli space of vector bundles on [Formula: see text]. With suitable hypotheses, we construct a fine moduli space [Formula: see text] for the vector bundles on [Formula: see text] whose restriction to [Formula: see text] belongs to [Formula: see text]. It is an affine bundle over the subvariety [Formula: see text] of bundles that can be extended to [Formula: see text]. In general this affine bundle is not banal. This applies in particular to Picard groups. We give also many new examples of primitive multiple schemes Y such that the dualizing sheaf [Formula: see text] is trivial.