5,597 research outputs found

### On the Hilbert scheme of curves in higher-dimensional projective space

In this paper we prove that, for any $n\ge 3$, there exist infinitely many
$r\in \N$ and for each of them a smooth, connected curve $C_r$ in $\P^r$ such
that $C_r$ lies on exactly $n$ 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

### Impurity in a bosonic Josephson junction: swallowtail loops, chaos, self-trapping and the poor man's Dicke model

We study a model describing $N$ identical bosonic atoms trapped in a
double-well potential together with a single impurity atom, comparing and
contrasting it throughout with the Dicke model. As the boson-impurity coupling
strength is varied, there is a symmetry-breaking pitchfork bifurcation which is
analogous to the quantum phase transition occurring in the Dicke model. Through
stability analysis around the bifurcation point, we show that the critical
value of the coupling strength has the same dependence on the parameters as the
critical coupling value in the Dicke model. We also show that, like the Dicke
model, the mean-field dynamics go from being regular to chaotic above the
bifurcation and macroscopic excitations of the bosons are observed. Overall,
the boson-impurity system behaves like a poor man's version of the Dicke model.Comment: 17 pages, 16 figure

### Dicke-type phase transition in a multimode optomechanical system

We consider the "membrane in the middle" optomechanical model consisting of a
laser pumped cavity which is divided in two by a flexible membrane that is
partially transmissive to light and subject to radiation pressure. Steady state
solutions at the mean-field level reveal that there is a critical strength of
the light-membrane coupling above which there is a symmetry breaking
bifurcation where the membrane spontaneously acquires a displacement either to
the left or the right. This bifurcation bears many of the signatures of a
second order phase transition and we compare and contrast it with that found in
the Dicke model. In particular, by studying limiting cases and deriving
dynamical critical exponents using the fidelity susceptibility method, we argue
that the two models share very similar critical behaviour. For example, the
obtained critical exponents indicate that they fall within the same
universality class. Away from the critical regime we identify, however, some
discrepancies between the two models. Our results are discussed in terms of
experimentally relevant parameters and we evaluate the prospects for realizing
Dicke-type physics in these systems.Comment: 14 pages, 6 figure

### Isomonodromic deformatiion with an irregular singularity and hyperelliptic curve

In this paper, we extend the result of Kitaev and Korotkin to the case where
a monodromy-preserving deformation has an irregular singularity. For the
monodromy-preserving deformation, we obtain the $\tau$-function whose
deformation parameters are the positions of regular singularities and the
parameter $t$ of an irregular singularity. Furthermore, the $\tau$-function is
expressed by the hyperelliptic $\Theta$ function moving the argument \z and
the period \B, where $t$ and the positions of regular singularities move $z$
and \B, respectively.Comment: 23 pages, 2 figure

### Elliptic fibrations associated with the Einstein spacetimes

Given a conformally nonflat Einstein spacetime we define a fibration $P$ over
it. The fibres of this fibration are elliptic curves (2-dimensional tori) or
their degenerate counterparts. Their topology depends on the algebraic type of
the Weyl tensor of the Einstein metric. The fibration $P$ is a double branched
cover of the bundle of null direction over the spacetime and is equipped with
six linearly independent 1-forms which satisfy certain relatively simple system
of equations.Comment: 15 pages, Late

### Quotients of E^n by A_{n+1} and Calabi-Yau manifolds

We give a simple construction, starting with any elliptic curve E, of an
n-dimensional Calabi-Yau variety of Kummer type (for any n>1), by considering
the quotient Y of the n-fold self-product of E by a natural action of the
alternating group A_{n+1} (in n+1 variables). The vanishing of H^m(Y, O_Y) for
0<m<n follows from the non-existence of (non-zero) fixed points in certain
representations of A_{n+1}. For n<4 we provide an explicit crepant resolution X
in characteristics different from 2,3. The key point is that Y can be realized
as a double cover of P^n branched along a hypersurface of degree 2(n+1).Comment: 9 page

### 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

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