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Search: MSC category 14J10 ( Families, moduli, classification: algebraic theory )

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1. CJM 2012 (vol 65 pp. 195)

Penegini, Matteo; Polizzi, Francesco
Surfaces with $p_g=q=2$, $K^2=6$, and Albanese Map of Degree $2$
We classify minimal surfaces of general type with $p_g=q=2$ and $K^2=6$ whose Albanese map is a generically finite double cover. We show that the corresponding moduli space is the disjoint union of three generically smooth irreducible components $\mathcal{M}_{Ia}$, $\mathcal{M}_{Ib}$, $\mathcal{M}_{II}$ of dimension $4$, $4$, $3$, respectively.

Keywords:surface of general type, abelian surface, Albanese map
Categories:14J29, 14J10

2. CJM 2010 (vol 62 pp. 1131)

Kleppe, Jan O.
Moduli Spaces of Reflexive Sheaves of Rank 2
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
Categories:14C05, qqqqq14D22, 14F05, 14J10, 14H50, 14B10, 13D02, 13D07

3. CJM 1997 (vol 49 pp. 675)

de Cataldo, Mark Andrea A.
Some adjunction-theoretic properties of codimension two non-singular subvarities of quadrics
We make precise the structure of the first two reduction morphisms associated with codimension two non-singular subvarieties of non-singular quadrics $\Q^n$, $n\geq 5$. We give a coarse classification of the same class of subvarieties when they are assumed not to be of log-general-type.}

Keywords:Adjunction Theory, classification, codimension two, conic bundles,, low codimension, non log-general-type, quadric, reduction, special, variety.
Categories:14C05, 14E05, 14E25, 14E30, 14E35, 14J10

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