Coherent Sheaves on Primitive Multiple Schemes

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 dim(X)+1. If n is the multiplicity of Y, then there is a canonical filtration X=X1⊂X2⊂⋯⊂Xn=Y such that Xi is a primitive multiple scheme of multiplicity i. The ideal sheaf IX of X is a line bundle on Xn−1, and 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 a generalization 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⟶IP⊗L⟶E⟶IP=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 supported 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.