Moduli spaces of reflexive sheaves of rank 2

TL;DR

This paper explores the deformation theory and moduli space structure of rank 2 reflexive sheaves on projective threefolds, establishing criteria for smoothness, unobstructedness, and component dimension without assuming Ext^2 vanishing.

Contribution

It introduces new criteria linking deformation functors of sheaves and subschemes, and characterizes smoothness and dimension of moduli spaces for reflexive sheaves with specific deficiency modules.

Findings

Criteria for smoothness and unobstructedness of moduli spaces.

Irreducible components containing diameter-one sheaves are generically smooth.

Provides bounds on the dimension of components with small deficiency modules.

Abstract

Let \sF be a coherent rank 2 sheaf on a scheme Y \subset \proj{n} of dimension at least two. In this paper we study the relationship between the functor which deforms a pair (\sF,\sigma), \sigma \in H^0(\sF), and the functor which deforms the corresponding pair (X,\xi) given as in the Serre correspondence. We prove that the scheme structure of e.g. the moduli scheme M_Y(P) of stable sheaves on a threefold Y at (\sF), and the scheme structure at (X) of the Hilbert scheme of curves on Y are closely related. Using this relationship we get criteria for the dimension and smoothness of M_Y(P) at (\sF), without assuming Ext^2(\sF,\sF) = 0. For reflexive sheaves on Y = \proj{3} whose deficiency module M = H_{*}^1(\sF) satisfies Ext^2(M,M) = 0 in degree zero (e.g. of diameter at most 2), we get necessary and sufficient conditions of unobstructedness which 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(\sF). It follows that every irreducible component of M_{\proj{3}}(P) containing a reflexive sheaf of diameter one is reduced (generically smooth). We also determine a good lower bound for the dimension of any component of M_{\proj{3}}(P) which contains a reflexive stable sheaf with "small" deficiency module M.