Sheaves on non-reduced curves in a projective surface

TL;DR

This paper investigates the moduli space of pure sheaves supported on non-reduced curves within a surface, providing dimension estimates and analyzing the fibers of the Hilbert-Chow morphism in specific geometric contexts.

Contribution

It offers new dimension estimates for the moduli stack of pure sheaves on non-reduced curves and studies the uniformity of fibers of the Hilbert-Chow morphism under certain conditions.

Findings

Dimension estimates for the moduli stack of pure sheaves on non-reduced curves.

The Hilbert-Chow morphism has fibers of equal dimension for Fano surfaces or surfaces with trivial canonical bundle.

Analysis applies when the linear system contains integral curves.

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 $nC~(n\geq 2)$ with $C$ an integral curve on $X$. We prove that the Hilbert-Chow morphism $h_{L,\chi}:\mathcal{M}^H_X(L,\chi)\rightarrow |L|$ 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.