Sheaves on non-reduced curves in a projective surface

Abstract

Sheaves on non-reduced curves can appear in moduli space of 1-dimensional semistable sheaves over a surface, and moduli space of Higgs bundles as well. We estimate the dimension of the stack ${\mathbf{M}}_X(nC,\chi )$ of pure sheaves supported at the non-reduced curve $nC(n\ge 2)$ with $C$ an integral curve on $X$. We prove that the Hilbert-Chow morphism $h_{L,\chi }:{\mathcal{M}}_X^H(L,\chi )\to |L|$ sending each semistable 1-dimensional sheaf to its support have all its fibers of the same dimension for $X$ Fano or with trivial canonical line bundle and $|L|$ contains integral curves.

arXiv comment18 pages