528 research outputs found
Remarks on endomorphisms and rational points
Let X be a variety over a number field and let f: X --> X be an "interesting"
rational self-map with a fixed point q. We make some general remarks concerning
the possibility of using the behaviour of f near q to produce many rational
points on X. As an application, we give a simplified proof of the potential
density of rational points on the variety of lines of a cubic fourfold
(originally obtained by Claire Voisin and the first author in 2007).Comment: LaTeX, 22 pages. v2: some minor observations added, misprints
corrected, appendix modified
Holomorphic symplectic geometry: a problem list
A list of open problems on holomorphic symplectic, contact and Poisson
manifolds
On p-adic lattices and Grassmannians
It is well-known that the coset spaces G(k((z)))/G(k[[z]]), for a reductive
group G over a field k, carry the geometric structure of an inductive limit of
projective k-schemes. This k-ind-scheme is known as the affine Grassmannian for
G. From the point of view of number theory it would be interesting to obtain an
analogous geometric interpretation of quotients of the form
G(W(k)[1/p])/G(W(k)), where p is a rational prime, W denotes the ring scheme of
p-typical Witt vectors, k is a perfect field of characteristic p and G is a
reductive group scheme over W(k). The present paper is an attempt to describe
which constructions carry over from the function field case to the p-adic case,
more precisely to the situation of the p-adic affine Grassmannian for the
special linear group G=SL_n. We start with a description of the R-valued points
of the p-adic affine Grassmannian for SL_n in terms of lattices over W(R),
where R is a perfect k-algebra. In order to obtain a link with geometry we
further construct projective k-subvarieties of the multigraded Hilbert scheme
which map equivariantly to the p-adic affine Grassmannian. The images of these
morphisms play the role of Schubert varieties in the p-adic setting. Further,
for any reduced k-algebra R these morphisms induce bijective maps between the
sets of R-valued points of the respective open orbits in the multigraded
Hilbert scheme and the corresponding Schubert cells of the p-adic affine
Grassmannian for SL_n.Comment: 36 pages. This is a thorough revision, in the form accepted by Math.
Zeitschrift, of the previously published preprint "On p-adic loop groups and
Grassmannians
Stable symmetries of plane sextics
We classify projective symmetries of irreducible plane sextics with simple
singularities which are stable under equivariant deformations. We also outline
a connection between order~2 stable symmetries and maximal trigonal curves
Multi-Hamiltonian structures for r-matrix systems
For the rational, elliptic and trigonometric r-matrices, we exhibit the links
between three "levels" of Poisson spaces: (a) Some finite-dimensional spaces of
matrix-valued holomorphic functions on the complex line; (b) Spaces of spectral
curves and sheaves supported on them; (c) Symmetric products of a surface. We
have, at each level, a linear space of compatible Poisson structures, and the
maps relating the levels are Poisson. This leads in a natural way to Nijenhuis
coordinates for these spaces. At level (b), there are Hamiltonian systems on
these spaces which are integrable for each Poisson structure in the family, and
which are such that the Lagrangian leaves are the intersections of the
symplective leaves over the Poisson structures in the family. Specific examples
include many of the well-known integrable systems.Comment: 26 pages, Plain Te
Fibrations on four-folds with trivial canonical bundles
Four-folds with trivial canonical bundles are divided into six classes
according to their holonomy group. We consider examples that are fibred by
abelian surfaces over the projective plane. We construct such fibrations in
five of the six classes, and prove that there is no such fibration in the sixth
class. We classify all such fibrations whose generic fibre is the Jacobian of a
genus two curve.Comment: 28 page
A template bank to search for gravitational waves from inspiralling compact binaries: II. Phenomenological model
Matched filtering is used to search for gravitational waves emitted by
inspiralling compact binaries in data from ground-based interferometers. One of
the key aspects of the detection process is the deployment of a set of
templates, also called a template bank, to cover the astrophysically
interesting region of the parameter space. In a companion paper, we described
the template-bank algorithm used in the analysis of LIGO data to search for
signals from non-spinning binaries made of neutron star and/or stellar-mass
black holes; this template bank is based upon physical template families. In
this paper, we describe the phenomenological template bank that was used to
search for gravitational waves from non-spinning black hole binaries (from
stellar mass formation) in the second, third and fourth LIGO science runs. We
briefly explain the design of the bank, whose templates are based on a
phenomenological detection template family. We show that this template bank
gives matches greater than 95% with the physical template families that are
expected to be captured by the phenomenological templates.Comment: 10 pages, 9 figure
Webs of Lagrangian Tori in Projective Symplectic Manifolds
For a Lagrangian torus A in a simply-connected projective symplectic manifold
M, we prove that M has a hypersurface disjoint from a deformation of A. This
implies that a Lagrangian torus in a compact hyperk\"ahler manifold is a fiber
of an almost holomorphic Lagrangian fibration, giving an affirmative answer to
a question of Beauville's. Our proof employs two different tools: the theory of
action-angle variables for algebraically completely integrable Hamiltonian
systems and Wielandt's theory of subnormal subgroups.Comment: 18 pages, minor latex problem fixe
Non-liftable Calabi-Yau spaces
We construct many new non-liftable three-dimensional Calabi-Yau spaces in
positive characteristic. The technique relies on lifting a nodal model to a
smooth rigid Calabi-Yau space over some number field as introduced by the first
author and D. van Straten.Comment: 16 pages, 5 tables; v2: minor corrections and addition
A comparison of methods for gravitational wave burst searches from LIGO and Virgo
The search procedure for burst gravitational waves has been studied using 24
hours of simulated data in a network of three interferometers (Hanford 4-km,
Livingston 4-km and Virgo 3-km are the example interferometers). Several
methods to detect burst events developed in the LIGO Scientific Collaboration
(LSC) and Virgo collaboration have been studied and compared. We have performed
coincidence analysis of the triggers obtained in the different interferometers
with and without simulated signals added to the data. The benefits of having
multiple interferometers of similar sensitivity are demonstrated by comparing
the detection performance of the joint coincidence analysis with LSC and Virgo
only burst searches. Adding Virgo to the LIGO detector network can increase by
50% the detection efficiency for this search. Another advantage of a joint
LIGO-Virgo network is the ability to reconstruct the source sky position. The
reconstruction accuracy depends on the timing measurement accuracy of the
events in each interferometer, and is displayed in this paper with a fixed
source position example.Comment: LIGO-Virgo working group submitted to PR
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