619 research outputs found
Codimension 3 Arithmetically Gorenstein Subschemes of projective -space
We study the lowest dimensional open case of the question whether every
arithmetically Cohen--Macaulay subscheme of is glicci, that is,
whether every zero-scheme in is glicci. We show that a set of points in general position in \PP^3 admits no strictly descending
Gorenstein liaison or biliaison. In order to prove this theorem, we establish a
number of important results about arithmetically Gorenstein zero-schemes in
.Comment: to appear in Annales de l'Institut Fourie
Hurwitz spaces of quadruple coverings of elliptic curves and the moduli space of abelian threefolds A_3(1,1,4)
We prove that the moduli space A_3(1,1,4) of polarized abelian threefolds
with polarization of type (1,1,4) is unirational. By a result of Birkenhake and
Lange this implies the unirationality of the isomorphic moduli space
A_3(1,4,4). The result is based on the study the Hurwitz space H_{4,n}(Y) of
quadruple coverings of an elliptic curve Y simply branched in n points. We
prove the unirationality of its codimension one subvariety H^{0}_{4,A}(Y) which
parametrizes quadruple coverings \pi:X --> Y with Tschirnhausen modules
isomorphic to A^{-1}, where A\in Pic^{n/2}Y, and for which \pi^*:J(Y)--> J(X)
is injective. This is an analog of the result of Arbarello and Cornalba that
the Hurwitz space H_{4,n}(P^1) is unirational.Comment: 28 pages, amslatex, to appear in Mathematische Nachrichte
Coherent analogues of matrix factorizations and relative singularity categories
We define the triangulated category of relative singularities of a closed
subscheme in a scheme. When the closed subscheme is a Cartier divisor, we
consider matrix factorizations of the related section of a line bundle, and
their analogues with locally free sheaves replaced by coherent ones. The
appropriate exotic derived category of coherent matrix factorizations is then
identified with the triangulated category of relative singularities, while the
similar exotic derived category of locally free matrix factorizations is its
full subcategory. The latter category is identified with the kernel of the
direct image functor corresponding to the closed embedding of the zero locus
and acting between the conventional (absolute) triangulated categories of
singularities. Similar results are obtained for matrix factorizations of
infinite rank; and two different "large" versions of the triangulated category
of relative singularities, corresponding to the approaches of Orlov and Krause,
are identified in the case of a Cartier divisor. A version of the
Thomason-Trobaugh-Neeman localization theory is proven for coherent matrix
factorizations and disproven for locally free matrix factorizations of finite
rank. Contravariant (coherent) and covariant (quasi-coherent) versions of the
Serre-Grothendieck duality theorems for matrix factorizations are established,
and pull-backs and push-forwards of matrix factorizations are discussed at
length. A number of general results about derived categories of the second kind
for CDG-modules over quasi-coherent CDG-algebras are proven on the way.
Hochschild (co)homology of matrix factorization categories are discussed in an
appendix.Comment: LaTeX 2e with pb-diagram and xy-pic; 114 pages, 13 commutative
diagrams. v.8: new sections 2.10, 3.1 and 3.7 inserted; v.9: appendix B
added, remarks inserted in sections 2.10 and 2.7, section 1.8 expanded; v.10:
new section 3.3 inserted, the whole paper has two authors now; v.11: small
corrections, additions, and improvements -- this is intended as the final
versio
Log canonical thresholds of Del Pezzo Surfaces in characteristic p
The global log canonical threshold of each non-singular complex del Pezzo
surface was computed by Cheltsov. The proof used Koll\'ar-Shokurov's
connectedness principle and other results relying on vanishing theorems of
Kodaira type, not known to be true in finite characteristic.
We compute the global log canonical threshold of non-singular del Pezzo
surfaces over an algebraically closed field. We give algebraic proofs of
results previously known only in characteristic . Instead of using of the
connectedness principle we introduce a new technique based on a classification
of curves of low degree. As an application we conclude that non-singular del
Pezzo surfaces in finite characteristic of degree lower or equal than are
K-semistable.Comment: 21 pages. Thorough rewrite following referee's suggestions. To be
published in Manuscripta Mathematic
A simple remark on a flat projective morphism with a Calabi-Yau fiber
If a K3 surface is a fiber of a flat projective morphisms over a connected
noetherian scheme over the complex number field, then any smooth connected
fiber is also a K3 surface. Observing this, Professor Nam-Hoon Lee asked if the
same is true for higher dimensional Calabi-Yau fibers. We shall give an
explicit negative answer to his question as well as a proof of his initial
observation.Comment: 8 pages, main theorem is generalized, one more remark is added,
mis-calculation and typos are corrected etc
Deformation theory of objects in homotopy and derived categories III: abelian categories
This is the third paper in a series. In part I we developed a deformation
theory of objects in homotopy and derived categories of DG categories. Here we
show how this theory can be used to study deformations of objects in homotopy
and derived categories of abelian categories. Then we consider examples from
(noncommutative) algebraic geometry. In particular, we study noncommutative
Grassmanians that are true noncommutative moduli spaces of structure sheaves of
projective subspaces in projective spaces.Comment: Alexander Efimov is a new co-author of this paper. Besides some minor
changes, a new part (part 3) about noncommutative Grassmanians was adde
Varieties of vacua in classical supersymmetric gauge theories
We give a simple description of the classical moduli space of vacua for
supersymmetric gauge theories with or without a superpotential. The key
ingredient in our analysis is the observation that the lagrangian is invariant
under the action of the complexified gauge group \Gc. From this point of view
the usual -flatness conditions are an artifact of Wess--Zumino gauge. By
using a gauge that preserves \Gc invariance we show that every constant
matter field configuration that extremizes the superpotential is \Gc
gauge-equivalent (in a sense that we make precise) to a unique classical
vacuum. This result is used to prove that in the absence of a superpotential
the classical moduli space is the algebraic variety described by the set of all
holomorphic gauge-invariant polynomials. When a superpotential is present, we
show that the classical moduli space is a variety defined by imposing
additional relations on the holomorphic polynomials. Many of these points are
already contained in the existing literature. The main contribution of the
present work is that we give a careful and self-contained treatment of limit
points and singularities.Comment: 14 pages, LaTeX (uses revtex.sty
A refined stable restriction theorem for vector bundles on quadric threefolds
Let E be a stable rank 2 vector bundle on a smooth quadric threefold Q in the
projective 4-space P. We show that the hyperplanes H in P for which the
restriction of E to the hyperplane section of Q by H is not stable form, in
general, a closed subset of codimension at least 2 of the dual projective
4-space, and we explicitly describe the bundles E which do not enjoy this
property. This refines a restriction theorem of Ein and Sols [Nagoya Math. J.
96, 11-22 (1984)] in the same way the main result of Coanda [J. reine angew.
Math. 428, 97-110 (1992)] refines the restriction theorem of Barth [Math. Ann.
226, 125-150 (1977)].Comment: Ann. Mat. Pura Appl. 201
Shapes of free resolutions over a local ring
We classify the possible shapes of minimal free resolutions over a regular
local ring. This illustrates the existence of free resolutions whose Betti
numbers behave in surprisingly pathological ways. We also give an asymptotic
characterization of the possible shapes of minimal free resolutions over
hypersurface rings. Our key new technique uses asymptotic arguments to study
formal Q-Betti sequences.Comment: 14 pages, 1 figure; v2: sections have been reorganized substantially
and exposition has been streamline
Introduction to Arithmetic Mirror Symmetry
We describe how to find period integrals and Picard-Fuchs differential
equations for certain one-parameter families of Calabi-Yau manifolds. These
families can be seen as varieties over a finite field, in which case we show in
an explicit example that the number of points of a generic element can be given
in terms of p-adic period integrals. We also discuss several approaches to
finding zeta functions of mirror manifolds and their factorizations. These
notes are based on lectures given at the Fields Institute during the thematic
program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physics
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