Generalized line bundles on primitive multiple curves and their moduli

In this paper, we study generalized line bundles over $C_n$, a primitive multiple curve of arbitrary multiplicity $n$, where $n$ is a positive integer. In particular, we give a structure theorem for them and we characterize their semistability in terms of $n-1$ integral invariants, the indices. These results are used to describe the irreducible components that contain stable generalized line bundles in the Simpson moduli space of semistable sheaves of generalized rank $n$ and fixed generalized degree on $C_n$. We compute also the dimension of the Zariski tangent space to this moduli space in a point representing a generic generalized line bundle (any generalized line bundle for $n=3$). In the case $n=1$ all the results are completely trivial, while the case $n=2$ has already been treated by Chen and Kass.