Coherent sheaves on primitive multiple curves

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 $I_X$ is the ideal sheaf of $X$ in $Y$ and $Y\not=X$, then $L=I_X/I_X^2$ 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 $P_{red,O_X(1)}(E)$ of a coherent sheaf $E$ on $Y$, depending on the choice of an ample line bundle $O_X(1)$ on $X$. If $E$ is a flat family of sheaves on $Y$ parameterized by a smooth curve $C$, then $P_{red,O_X(1)}(E_c)$ does not depend on $c\in C$. 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 $$0\longrightarrow I_P\otimes L\longrightarrow E\longrightarrow I_P=E_{|X} \longrightarrow 0 \ ,$$ where $I_P\subset O_X$ is the ideal sheaf of a point $P\in X$. They can also be described as the ideal sheaves $E$ of subschemes of $Y$ concentrated on $P$, and such that $E_P$ is generated by two elements whose images in $O_{X,P}$ generate the maximal ideal. There is a moduli space for such sheaves, which is an affine bundle on $X$ with associated vector bundle $T_X\otimes L$ (where $T_X$ is the tangent bundle of $X$). The associated class in $H^1(X,T_X\otimes L)$ can be determined.