Moduli of vector bundles on primitive multiple schemes

A primitive multiple scheme is a Cohen-Macaulay $Y$ such that the associated reduced $X=Y_{red}$ smooth, irreducible, and can be locally embedded in smooth variety of dimension $\dim(X)+1$. If $n$ multiplicity $Y$, there canonical filtration $X=X_1\subset X_2\subset\cdots\subset X_n=Y$, $X_i$ $i$. The simplest example trivial to line bundle $L$ on $X$: it $n$-th infinitesimal neighborhood $X$, $L^*$ by zero section. main subject this paper construction properties fine moduli spaces vector bundles schemes. Suppose $Y=X_n$ $n$, extended $X_{n+1}$ $n+1$, let $M_n$ space $X_n$. With suitable hypotheses, we construct $M_{n+1}$ for whose restriction $X_n$ belongs $M_n$. It an affine over subvariety $N_n\subset M_n$ $X_{n+1}$. In general not banal. This applies particular Picard groups. We give also many new examples schemes dualizing sheaf $\omega_Y$ trivial.