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# Moduli Spaces of Reflexive Sheaves of Rank 2

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http://dx.doi.org/10.4153/CJM-2010-057-6
Canad. J. Math. 62(2010), 1131-1154
Published:2010-07-06
Printed: Oct 2010
• Jan O. Kleppe,
Oslo University College, Faculty of Engineering , Pb. 4 St. Olavs plass, 0130, Oslo, Norway
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## Abstract

Let $\mathcal{F}$ be a coherent rank $2$ sheaf on a scheme $Y \subset \mathbb{P}^{n}$ of dimension at least two and let $X \subset Y$ be the zero set of a section $\sigma \in H^0(\mathcal{F})$. In this paper, we study the relationship between the functor that deforms the pair $(\mathcal{F},\sigma)$ and the two functors that deform $\mathcal{F}$ 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 \emph{e.g.,} the moduli scheme ${\rm M_Y}(P)$ of stable sheaves on a threefold $Y$ at $(\mathcal{F})$, 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 ${\rm M_{Y}}(P)$ at $(\mathcal{F})$, without assuming ${\textrm{Ext}^2}(\mathcal{F} ,\mathcal{F} ) = 0$. For reflexive sheaves on $Y=\mathbb{P}^{3}$ whose deficiency module $M = H_{*}^1(\mathcal{F})$ satisfies ${_{0}\! \textrm{Ext}^2}(M ,M ) = 0$ (\emph{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 $H_{*}^0(\mathcal{F})$. Moreover, we show that every irreducible component of ${\rm M}_{\mathbb{P}^{3}}(P)$ 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 ${\rm M}_{\mathbb{P}^{3}}(P)$ that contains a reflexive stable sheaf with small'' deficiency module $M$.

Keywords:  moduli space, reflexive sheaf, Hilbert scheme, space curve, Buchsbaum sheaf, unobstructedness, cup product, graded Betti numbers.xdvi

MSC Classifications:  14C05 - Parametrization (Chow and Hilbert schemes) qqqqq14D22 - unknown classification qqqqq14D2214F05 - Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 14J10 - Families, moduli, classification: algebraic theory 14H50 - Plane and space curves 14B10 - Infinitesimal methods [See also 13D10] 13D02 - Syzygies, resolutions, complexes 13D07 - Homological functors on modules (Tor, Ext, etc.)

© Canadian Mathematical Society, 2013

© Canadian Mathematical Society, 2013 : http://www.cms.math.ca/
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