59 research outputs found
Simply connected projective manifolds in characteristic have no nontrivial stratified bundles
We show that simply connected projective manifolds in characteristic
have no nontrivial stratified bundles. This gives a positive answer to a
conjecture by D. Gieseker. The proof uses Hrushovski's theorem on periodic
points.Comment: 16 pages. Revised version contains a more general theorem on torsion
points on moduli, together with an illustration in rank 2 due to M. Raynaud.
Reference added. Last version has some typos corrected. Appears in
Invent.math
Harper operators, Fermi curves, and Picard-Fuchs equations
This paper is a continuation of the work on the spectral problem of Harper
operator using algebraic geometry. We continue to discuss the local monodromy
of algebraic Fermi curves based on Picard-Lefschetz formula. The density of
states over approximating components of Fermi curves satisfies a Picard-Fuchs
equation. By the property of Landen transformation, the density of states has a
Lambert series as the quarter period. A -expansion of the energy level can
be derived from a mirror map as in the B-model.Comment: v2, 13 pages, minor changes have been mad
Elliptic curve configurations on Fano surfaces
The elliptic curves on a surface of general type constitute an obstruction
for the cotangent sheaf to be ample. In this paper, we give the classification
of the configurations of the elliptic curves on the Fano surface of a smooth
cubic threefold. That means that we give the number of such curves, their
intersections and a plane model. This classification is linked to the
classification of the automorphism groups of theses surfaces.Comment: 17 pages, accepted and shortened version, the rest will appear in
"Fano surfaces with 12 or 30 elliptic curves
Geometric invariant theory of syzygies, with applications to moduli spaces
We define syzygy points of projective schemes, and introduce a program of
studying their GIT stability. Then we describe two cases where we have managed
to make some progress in this program, that of polarized K3 surfaces of odd
genus, and of genus six canonical curves. Applications of our results include
effectivity statements for divisor classes on the moduli space of odd genus K3
surfaces, and a new construction in the Hassett-Keel program for the moduli
space of genus six curves.Comment: v1: 23 pages, submitted to the Proceedings of the Abel Symposium
2017, v2: final version, corrects a sign error and resulting divisor class
calculations on the moduli space of K3 surfaces in Section 5, other minor
changes, In: Christophersen J., Ranestad K. (eds) Geometry of Moduli.
Abelsymposium 2017. Abel Symposia, vol 14. Springer, Cha
Higher Order Potential Expansion for the Continuous Limits of the Toda Hierarchy
A method for introducing the higher order terms in the potential expansion to
study the continuous limits of the Toda hierarchy is proposed in this paper.
The method ensures that the higher order terms are differential polynomials of
the lower ones and can be continued to be performed indefinitly. By introducing
the higher order terms, the fewer equations in the Toda hierarchy are needed in
the so-called recombination method to recover the KdV hierarchy. It is shown
that the Lax pairs, the Poisson tensors, and the Hamiltonians of the Toda
hierarchy tend towards the corresponding ones of the KdV hierarchy in
continuous limit.Comment: 20 pages, Latex, to be published in Journal of Physics
Scattering theory for lattice operators in dimension
This paper analyzes the scattering theory for periodic tight-binding
Hamiltonians perturbed by a finite range impurity. The classical energy
gradient flow is used to construct a conjugate (or dilation) operator to the
unperturbed Hamiltonian. For dimension the wave operator is given by
an explicit formula in terms of this dilation operator, the free resolvent and
the perturbation. From this formula the scattering and time delay operators can
be read off. Using the index theorem approach, a Levinson theorem is proved
which also holds in presence of embedded eigenvalues and threshold
singularities.Comment: Minor errors and misprints corrected; new result on absense of
embedded eigenvalues for potential scattering; to appear in RM
On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators
The article is devoted to the following question. Consider a periodic
self-adjoint difference (differential) operator on a graph (quantum graph) G
with a co-compact free action of the integer lattice Z^n. It is known that a
local perturbation of the operator might embed an eigenvalue into the
continuous spectrum (a feature uncommon for periodic elliptic operators of
second order). In all known constructions of such examples, the corresponding
eigenfunction is compactly supported. One wonders whether this must always be
the case. The paper answers this question affirmatively. What is more
surprising, one can estimate that the eigenmode must be localized not far away
from the perturbation (in a neighborhood of the perturbation's support, the
width of the neighborhood determined by the unperturbed operator only).
The validity of this result requires the condition of irreducibility of the
Fermi (Floquet) surface of the periodic operator, which is expected to be
satisfied for instance for periodic Schroedinger operators.Comment: Submitted for publicatio
A functorial construction of moduli of sheaves
We show how natural functors from the category of coherent sheaves on a
projective scheme to categories of Kronecker modules can be used to construct
moduli spaces of semistable sheaves. This construction simplifies or clarifies
technical aspects of existing constructions and yields new simpler definitions
of theta functions, about which more complete results can be proved.Comment: 52 pp. Dedicated to the memory of Joseph Le Potier. To appear in
Inventiones Mathematicae. Slight change in the definition of the Kronecker
algebra in Secs 1 (p3) and 2.2 (p6), with corresponding small alterations
elsewhere, to make the constructions work for non-reduced schemes. Section
6.5 rewritten. Remark 2.6 and new references adde
The classification of isotrivially fibred surfaces with p_g=q=2
An isotrivially fibred surface is a smooth projective surface endowed with a
morphism onto a curve such that all the smooth fibres are isomorphic to each
other. The first goal of this paper is to classify the isotrivially fibred
surfaces with completing and extending a result of Zucconi. As an
important byproduct, we provide new examples of minimal surfaces of general
type with and and a first example with .Comment: Main paper by M.Penegini. Appendix by S.Rollenske. 31 pages, 6
Figures. v2 changed group relations in Theorem 5.2, changes in Theorem 5.7,
new proof of Theorem 4.15, minor corrections of misprint
Boundary Fixed Points, Enhanced Gauge Symmetry and Singular Bundles on K3
We investigate certain fixed points in the boundary conformal field theory
representation of type IIA D-branes on Gepner points of K3. They correspond
geometrically to degenerate brane configurations, and physically lead to
enhanced gauge symmetries on the world-volume. Non-abelian gauge groups arise
if the stabilizer group of the fixed points is realized projectively, which is
similar to D-branes on orbifolds with discrete torsion. Moreover, the fixed
point boundary states can be resolved into several irreducible components.
These correspond to bound states at threshold and can be viewed as (non-locally
free) sub-sheaves of semi-stable sheaves. Thus, the BCFT fixed points appear to
carry two-fold geometrical information: on the one hand they probe the boundary
of the instanton moduli space on K3, on the other hand they probe discrete
torsion in D-geometry.Comment: harvmac, 20
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