4,360 research outputs found
On the Hilbert scheme of curves in higher-dimensional projective space
In this paper we prove that, for any , there exist infinitely many
and for each of them a smooth, connected curve in such
that lies on exactly irreducible components of the Hilbert scheme
\hilb(\P^r). This is proven by reducing the problem to an analogous statement
for the moduli of surfaces of general type.Comment: latex, 12 pages, no figure
Geodesic Completeness for Sobolev Metrics on the Space of Immersed Plane Curves
We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as initial data in certain Sobolev spaces. Thus the space of closed plane curves equipped with such a metric is geodesically complete. We find lower bounds for the geodesic distance in terms of curvature and its derivatives
On compatibility between isogenies and polarisations of abelian varieties
We discuss the notion of polarised isogenies of abelian varieties, that is,
isogenies which are compatible with given principal polarisations. This is
motivated by problems of unlikely intersections in Shimura varieties. Our aim
is to show that certain questions about polarised isogenies can be reduced to
questions about unpolarised isogenies or vice versa.
Our main theorem concerns abelian varieties B which are isogenous to a fixed
abelian variety A. It establishes the existence of a polarised isogeny A to B
whose degree is polynomially bounded in n, if there exist both an unpolarised
isogeny A to B of degree n and a polarised isogeny A to B of unknown degree. As
a further result, we prove that given any two principally polarised abelian
varieties related by an unpolarised isogeny, there exists a polarised isogeny
between their fourth powers.
The proofs of both theorems involve calculations in the endomorphism algebras
of the abelian varieties, using the Albert classification of these endomorphism
algebras and the classification of Hermitian forms over division algebras
Alternating groups and moduli space lifting Invariants
Main Theorem: Spaces of r-branch point 3-cycle covers, degree n or Galois of
degree n!/2 have one (resp. two) component(s) if r=n-1 (resp. r\ge n). Improves
Fried-Serre on deciding when sphere covers with odd-order branching lift to
unramified Spin covers. We produce Hurwitz-Torelli automorphic functions on
Hurwitz spaces, and draw Inverse Galois conclusions. Example: Absolute spaces
of 3-cycle covers with +1 (resp. -1) lift invariant carry canonical even (resp.
odd) theta functions when r is even (resp. odd). For inner spaces the result is
independent of r. Another use appears in,
http://www.math.uci.edu/~mfried/paplist-mt/twoorbit.html, "Connectedness of
families of sphere covers of A_n-Type." This shows the M(odular) T(ower)s for
the prime p=2 lying over Hurwitz spaces first studied by,
http://www.math.uci.edu/~mfried/othlist-cov/hurwitzLiu-Oss.pdf, Liu and
Osserman have 2-cusps. That is sufficient to establish the Main Conjecture: (*)
High tower levels are general-type varieties and have no rational points.For
infinitely many of those MTs, the tree of cusps contains a subtree -- a spire
-- isomorphic to the tree of cusps on a modular curve tower. This makes
plausible a version of Serre's O(pen) I(mage) T(heorem) on such MTs.
Establishing these modular curve-like properties opens, to MTs, modular
curve-like thinking where modular curves have never gone before. A fuller html
description of this paper is at
http://www.math.uci.edu/~mfried/paplist-cov/hf-can0611591.html .Comment: To appear in the Israel Journal as of 1/5/09; v4 is corrected from
proof sheets, but does include some proof simplification in \S
Ultradiscretization of the solution of periodic Toda equation
A periodic box-ball system (pBBS) is obtained by ultradiscretizing the
periodic discrete Toda equation (pd Toda eq.). We show the relation between a
Young diagram of the pBBS and a spectral curve of the pd Toda eq.. The formula
for the fundamental cycle of the pBBS is obtained as a colloraly.Comment: 41 pages; 7 figure
Notes on Euclidean Wilson loops and Riemann Theta functions
The AdS/CFT correspondence relates Wilson loops in N=4 SYM theory to minimal
area surfaces in AdS5 space. In this paper we consider the case of Euclidean
flat Wilson loops which are related to minimal area surfaces in Euclidean AdS3
space. Using known mathematical results for such minimal area surfaces we
describe an infinite parameter family of analytic solutions for closed Wilson
loops. The solutions are given in terms of Riemann theta functions and the
validity of the equations of motion is proven based on the trisecant identity.
The world-sheet has the topology of a disk and the renormalized area is written
as a finite, one-dimensional contour integral over the world-sheet boundary. An
example is discussed in detail with plots of the corresponding surfaces.
Further, for each Wilson loops we explicitly construct a one parameter family
of deformations that preserve the area. The parameter is the so called spectral
parameter. Finally, for genus three we find a map between these Wilson loops
and closed curves inside the Riemann surface.Comment: 35 pages, 7 figures, pdflatex. V2: References added. Typos corrected.
Some points clarifie
Semiclassical Strings in AdS_5 x S^5 and Automorphic Functions
Using AdS/CFT we derive from the folded spinning string ordinary differential
equations for the anomalous dimension of the dual N=4 SYM twist-two operators
at strong coupling. We show that for large spin the asymptotic solutions have
the Gribov-Lipatov recirocity property. To obtain this result we use a hidden
modular invariance of the energy-spin relation of the folded spinning string.
Further we identify the Moch-Vermaseren-Vogt (MVV) relations, which were first
recognized in plain QCD calculations, as the recurrence relations of the
asymptotic series ansatz.Comment: 4 page
On a new compactification of moduli of vector bundles on a surface, IV: Nonreduced moduli
The construction for nonreduced projective moduli scheme of semistable
admissible pairs is performed. We establish the relation of this moduli scheme
with reduced moduli scheme built up in the previous article and prove that
nonreduced moduli scheme contains an open subscheme which is isomorphic to
moduli scheme of semistable vector bundles.Comment: 20 pages, additions and removal
The Bethe ansatz in a periodic box-ball system and the ultradiscrete Riemann theta function
Vertex models with quantum group symmetry give rise to integrable cellular
automata at q=0. We study a prototype example known as the periodic box-ball
system. The initial value problem is solved in terms of an ultradiscrete
analogue of the Riemann theta function whose period matrix originates in the
Bethe ansatz at q=0.Comment: 11 pages, 1 figur
Non-Abelian adiabatic statistics and Hall viscosity in quantum Hall states and p_x+ip_y paired superfluids
Many trial wavefunctions for fractional quantum Hall states in a single
Landau level are given by functions called conformal blocks, taken from some
conformal field theory. Also, wavefunctions for certain paired states of
fermions in two dimensions, such as p_x+ip_y states, reduce to such a form at
long distances. Here we investigate the adiabatic transport of such
many-particle trial wavefunctions using methods from two-dimensional field
theory. One context for this is to calculate the statistics of widely-separated
quasiholes, which has been predicted to be non-Abelian in a variety of cases.
The Berry phase or matrix (holonomy) resulting from adiabatic transport around
a closed loop in parameter space is the same as the effect of analytic
continuation around the same loop with the particle coordinates held fixed
(monodromy), provided the trial functions are orthonormal and holomorphic in
the parameters so that the Berry vector potential (or connection) vanishes. We
show that this is the case (up to a simple area term) for paired states
(including the Moore-Read quantum Hall state), and present general conditions
for it to hold for other trial states (such as the Read-Rezayi series). We
argue that trial states based on a non-unitary conformal field theory do not
describe a gapped topological phase, at least in many cases. By considering
adiabatic variation of the aspect ratio of the torus, we calculate the Hall
viscosity, a non-dissipative viscosity coefficient analogous to Hall
conductivity, for paired states, Laughlin states, and more general quantum Hall
states. Hall viscosity is an invariant within a topological phase, and is
generally proportional to the "conformal spin density" in the ground state.Comment: 44 pages, RevTeX; v2 minor changes; v3 typos corrected, three small
addition
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