Sheaves on non-reduced curves in a projective surface

Abstract

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