3,321 research outputs found
Self-dual projective toric varieties
Let T be a torus over an algebraically closed field k of characteristic 0,
and consider a projective T-module P(V). We determine when a projective toric
subvariety X of P(V) is self-dual, in terms of the configuration of weights of
V.Comment: 26 pages, 1 figure. Minor change
Higher order selfdual toric varieties
The notion of higher order dual varieties of a projective variety, introduced
in \cite{P83}, is a natural generalization of the classical notion of
projective duality. In this paper we present geometric and combinatorial
characterizations of those equivariant projective toric embeddings that satisfy
higher order selfduality. We also give several examples and general
constructions. In particular, we highlight the relation with Cayley-Bacharach
questions and with Cayley configurations.Comment: 21 page
A Simple Combinatorial Criterion for Projective Toric Manifolds with Dual Defect
We show that any smooth lattice polytope P with codegree greater or equal
than (dim(P)+3)/2 (or equivalently, with degree smaller than dim(P)/2), defines
a dual defective projective toric manifold. This implies that P is Q-normal (in
the terminology of a recent paper by Di Rocco, Piene and the first author) and
answers partially an adjunction-theoretic conjecture by Beltrametti and
Sommese. Also, it follows that smooth lattice polytopes with this property are
precisely strict Cayley polytopes, which completes the answer of a question of
Batyrev and the second author in the nonsingular case.Comment: 12 page
Crystals, instantons and quantum toric geometry
We describe the statistical mechanics of a melting crystal in three
dimensions and its relation to a diverse range of models arising in
combinatorics, algebraic geometry, integrable systems, low-dimensional gauge
theories, topological string theory and quantum gravity. Its partition function
can be computed by enumerating the contributions from noncommutative instantons
to a six-dimensional cohomological gauge theory, which yields a dynamical
realization of the crystal as a discretization of spacetime at the Planck
scale. We describe analogous relations between a melting crystal model in two
dimensions and N=4 supersymmetric Yang-Mills theory in four dimensions. We
elaborate on some mathematical details of the construction of the quantum
geometry which combines methods from toric geometry, isospectral deformation
theory and noncommutative geometry in braided monoidal categories. In
particular, we relate the construction of noncommutative instantons to deformed
ADHM data, torsion-free modules and a noncommutative twistor correspondence.Comment: 33 pages, 5 figures; Contribution to the proceedings of "Geometry and
Physics in Cracow", Jagiellonian University, Cracow, Poland, September 21-25,
2010. To be published in Acta Physica Polonica Proceedings Supplemen
Positivity of the diagonal
We study how the geometry of a projective variety is reflected in the
positivity properties of the diagonal considered as a cycle on . We analyze when the diagonal is big, when it is nef, and when it is
rigid. In each case, we give several implications for the geometric properties
of . For example, when the diagonal is big, we prove that the Hodge groups
vanish for . We also classify varieties of low dimension
where the diagonal is nef and big.Comment: 23 pages; v2: updated attributions and minor change
Algebraic deformations of toric varieties II. Noncommutative instantons
We continue our study of the noncommutative algebraic and differential
geometry of a particular class of deformations of toric varieties, focusing on
aspects pertinent to the construction and enumeration of noncommutative
instantons on these varieties. We develop a noncommutative version of twistor
theory, which introduces a new example of a noncommutative four-sphere. We
develop a braided version of the ADHM construction and show that it
parametrizes a certain moduli space of framed torsion free sheaves on a
noncommutative projective plane. We use these constructions to explicitly build
instanton gauge bundles with canonical connections on the noncommutative
four-sphere that satisfy appropriate anti-selfduality equations. We construct
projective moduli spaces for the torsion free sheaves and demonstrate that they
are smooth. We define equivariant partition functions of these moduli spaces,
finding that they coincide with the usual instanton partition functions for
supersymmetric gauge theories on C^2.Comment: 62 pages; v2: typos corrected, references updated; Final version to
be published in Advances in Theoretical and Mathematical Physic
Gale duality and Koszul duality
Given an affine hyperplane arrangement with some additional structure, we
define two finite-dimensional, noncommutative algebras, both of which are
motivated by the geometry of hypertoric varieties. We show that these algebras
are Koszul dual to each other, and that the roles of the two algebras are
reversed by Gale duality. We also study the centers and representation
categories of our algebras, which are in many ways analogous to integral blocks
of category O.Comment: 55 pages; v2 contains significant revisions to proofs and to some of
the results. Section 7 has been deleted; that material will be incorporated
into a later paper by the same author
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