Coherent sheaves on primitive multiple curves

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 (X)+1. If IX is the ideal sheaf of X in Y and Y=X, then L=IX/IX² is a line bundle on X, called the associated line bundle of Y. Even if X is projective, Y needs not to be quasi projective. We define in every case the reduced Hilbert polynomial Pred,OX(1)(E) of a coherent sheaf E on Y, depending on the choice of an ample line bundle OX(1) on X. If E is a flat family of sheaves on Y parameterized by a smooth curve C, then Pred,OX(1)(Ec) does not depend on c∈ C. We study flat families of sheaves in two important cases: the families of quasi locally free sheaves, and if $n=2$ those of balanced sheaves. Balanced sheaves are generalizations of vector bundles on Y, and could be used to expand already known moduli spaces of vector bundles on Y. When X is a smooth projective surface, and Y is of multiplicity 2 we study the simplest examples of balanced sheaves: the sheaves E such that there is an exact sequence \[0 I_P⊗ L E I_P=E|X 0 \ , \] where IP⊂ OX is the ideal sheaf of a point P∈ X. They can also be described as the ideal sheaves E of subschemes of Y concentrated on P, and such that EP is generated by two elements whose images in OX,P generate the maximal ideal. There is a moduli space for such sheaves, which is an affine bundle on X with associated vector bundle TX⊗ L (where TX is the tangent bundle of X). The associated class in H¹(X,TX⊗ L) can be determined.