26 research outputs found
Noncommutative plane curves
In this paper we study noncommutative plane curves, i.e. non-commutative
k-algebras for which the 1-dimensional simple modules form a plane curve. We
study extensions of simple modules and we try to enlighten the completion
problem, i.e. understanding the connection between simple modules of different
dimension.Comment: 31 page
Deformation theory of objects in homotopy and derived categories I: general theory
This is the first paper in a series. We develop a general deformation theory
of objects in homotopy and derived categories of DG categories. Namely, for a
DG module over a DG category we define four deformation functors \Def
^{\h}(E), \coDef ^{\h}(E), \Def (E), \coDef (E). The first two functors
describe the deformations (and co-deformations) of in the homotopy
category, and the last two - in the derived category. We study their properties
and relations. These functors are defined on the category of artinian (not
necessarily commutative) DG algebras.Comment: Alexander Efimov is a new co-author of this paper. Besides some minor
changes, Proposition 7.1 and Theorem 8.1 were correcte
Moduli spaces of reflexive sheaves of rank 2
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.Comment: 19 page
Deformations of modules of maximal grade and the Hilbert scheme at determinantal schemes
Let R be a polynomial ring and M a finitely generated graded R-module of
maximal grade (which means that the ideal I_t(\cA) generated by the maximal
minors of a homogeneous presentation matrix, \cA, of M has maximal codimension
in R). Suppose X:=Proj(R/I_t(\cA)) is smooth in a sufficiently large open
subset and dim X > 0. Then we prove that the local graded deformation functor
of M is isomorphic to the local Hilbert (scheme) functor at X \subset Proj(R)
under a week assumption which holds if dim X > 1. Under this assumptions we get
that the Hilbert scheme is smooth at (X), and we give an explicit formula for
the dimension of its local ring. As a corollary we prove a conjecture of R. M.
Mir\'o-Roig and the author that the closure of the locus of standard
determinantal schemes with fixed degrees of the entries in a presentation
matrix is a generically smooth component V of the Hilbert scheme. Also their
conjecture on the dimension of V is proved for dim X > 0. The cohomology
H^i_{*}({\cN}_X) of the normal sheaf of X in Proj(R) is shown to vanish for 0 <
i < dim X-1. Finally the mentioned results, slightly adapted, remain true
replacing R by any Cohen-Macaulay quotient of a polynomial ring.Comment: 24 page
Pasquale del Pezzo, Duke of Caianello, Neapolitan mathematician
This article is dedicated to a reconstruction of some events and achievements, both personal and scientific, in the life of the Neapolitan mathematician Pasquale del Pezzo, Duke of Caianello