Moduli Spaces of Reflexive Sheaves of Rank 2

Abstract

AbstractLet ℱ be a coherent rank 2 sheaf on a scheme Y ⊂ ℙn of dimension at least two and let X ⊂ Y be the zero set of a section σ ∈ H0(ℱ). In this paper, we study the relationship between the functor that deforms the pair (ℱ, σ) and the two functors that deform F on Y, and X in Y, respectively. By imposing some conditions on two forgetful maps between the functors, we prove that the scheme structure of e.g., the moduli scheme MY(P) of stable sheaves on a threefold Y at (ℱ), and the scheme structure at (X) of the Hilbert scheme of curves on Y become closely related. Using this relationship, we get criteria for the dimension and smoothness of MY(P) at (ℱ), without assuming Ext2(ℱ,ℱ) = 0. For reflexive sheaves on Y = ℙ3 whose deficiencymodule M = H*1 (ℱ) satisfies 0Ext2(M,M) = 0 (e.g., of diameter at most 2), we get necessary and sufficient conditions of unobstructedness that coincide in the diameter one case. The conditions are further equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of H*0 (ℱ). Moreover, we show that every irreducible component of Mℙ3 (P) containing a reflexive sheaf of diameter one is reduced (generically smooth) and we compute its dimension. We also determine a good lower bound for the dimension of any component of Mℙ3 (P) that contains a reflexive stable sheaf with “small” deficiency module M.