Seminar on Moduli Spaces and Related Topics | 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 M_X(nC, \chi) 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} : M_X^H(L, \chi) -> |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. The strategy is to firstly deal with the case with C smooth and then do induction on the arithmetic genus of C which once can decrease by a blow-up given C singular. As an application, we generalize the result of Maulik-Shen on the cohomology \chi-independence of M_X^H(L,\chi) to X any del Pezzo surface not necessarily toric.