MODULI OF VECTOR BUNDLES ON PRIMITIVE MULTIPLE SCHEMES
Résumé
Resume. A primitive multiple scheme is a Cohen-Macaulay scheme Y such that the associated reduced scheme X = Y red is smooth, irreducible, and that Y can be locally embedded in a smooth variety of dimension dim(X) + 1. If n is the multiplicity of Y , there is a canonical filtration X = X 1 ⊂ X 2 ⊂ • • • ⊂ X n = Y , such that X i 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 n-th infinitesimal neighborhood of X, embedded in the line bundle L * 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 Y = X n is of multiplicity n, and can be extended to X n+1 of multiplicity n + 1, and let M n a fine moduli space of vector bundles on X n. With suitable hypotheses, we construct a fine moduli space M n+1 for the vector bundles on X n+1 whose restriction to X n belongs to M n. It is an affine bundle over the subvariety N n ⊂ M n of bundles that can be extended to X n+1. 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 ω Y is trivial (in the case of surfaces they are called K3-carpets).
Domaines
Géométrie algébrique [math.AG]
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