Smooth rationally connected threefolds contain all smooth curves

We show that if X is a smooth rationally connected threefold and C is a smooth projective curve then C can be embedded in X. Furthermore, a version of this property characterises rationally connected varieties of dimension at least 3. We give some details about the toric case.Comment: Version 1 was called "Any smooth toric threefold contains all curves". This version is completely rewritten and proves a much stronger result, following suggestions of Janos Kolla

Open orbifold Gromov-Witten invariants of [C^3/Z_n]: localization and mirror symmetry

We develop a mathematical framework for the computation of open orbifold Gromov-Witten invariants of [C^3/Z_n], and provide extensive checks with predictions from open string mirror symmetry. To this aim we set up a computation of open string invariants in the spirit of Katz-Liu, defining them by localization. The orbifold is viewed as an open chart of a global quotient of the resolved conifold, and the Lagrangian as the fixed locus of an appropriate anti-holomorphic involution. We consider two main applications of the formalism. After warming up with the simpler example of [C^3/Z_3], where we verify physical predictions of Bouchard, Klemm, Marino and Pasquetti, the main object of our study is the richer case of [C^3/Z_4], where two different choices are allowed for the Lagrangian. For one choice, we make numerical checks to confirm the B-model predictions; for the other, we prove a mirror theorem for orbifold disc invariants, match a large number of annulus invariants, and give mirror symmetry predictions for open string invariants of genus \leq 2.Comment: 44 pages + appendices; v2: exposition improved, misprints corrected, version to appear on Selecta Mathematica; v3: last minute mistake found and fixed for the symmetric brane setup of [C^3/Z_4]; in pres

Seidel elements and mirror transformations

The goal of this article is to give a precise relation between the mirror symmetry transformation of Givental and the Seidel elements for a smooth projective toric variety $X$ with $-K_X$ nef. We show that the Seidel elements entirely determine the mirror transformation and mirror coordinates.Comment: 36 pages. We corrected several issues as pointed out by the refere

Prepotentials for local mirror symmetry via Calabi-Yau fourfolds

In this paper, we first derive an intrinsic definition of classical triple intersection numbers of K_S, where S is a complex toric surface, and use this to compute the extended Picard-Fuchs system of K_S of our previous paper, without making use of the instanton expansion. We then extend this formalism to local fourfolds K_X, where X is a complex 3-fold. As a result, we are able to fix the prepotential of local Calabi-Yau threefolds K_S up to polynomial terms of degree 2. We then outline methods of extending the procedure to non canonical bundle cases.Comment: 42 pages, 7 figures. Expanded, reorganized, and added a theoretical background for the calculation

The symplectic Deligne-Mumford stack associated to a stacky polytope

We discuss a symplectic counterpart of the theory of stacky fans. First, we define a stacky polytope and construct the symplectic Deligne-Mumford stack associated to the stacky polytope. Then we establish a relation between stacky polytopes and stacky fans: the stack associated to a stacky polytope is equivalent to the stack associated to a stacky fan if the stacky fan corresponds to the stacky polytope.Comment: 20 pages; v2: To appear in Results in Mathematic

Encoding points on hyperelliptic curves over finite fields in deterministic polynomial time

We present families of (hyper)elliptic curve which admit an efficient deterministic encoding function

Orbit Determination with the two-body Integrals

We investigate a method to compute a finite set of preliminary orbits for solar system bodies using the first integrals of the Kepler problem. This method is thought for the applications to the modern sets of astrometric observations, where often the information contained in the observations allows only to compute, by interpolation, two angular positions of the observed body and their time derivatives at a given epoch; we call this set of data attributable. Given two attributables of the same body at two different epochs we can use the energy and angular momentum integrals of the two-body problem to write a system of polynomial equations for the topocentric distance and the radial velocity at the two epochs. We define two different algorithms for the computation of the solutions, based on different ways to perform elimination of variables and obtain a univariate polynomial. Moreover we use the redundancy of the data to test the hypothesis that two attributables belong to the same body (linkage problem). It is also possible to compute a covariance matrix, describing the uncertainty of the preliminary orbits which results from the observation error statistics. The performance of this method has been investigated by using a large set of simulated observations of the Pan-STARRS project.Comment: 23 pages, 1 figur

Microwave properties of (PrxY1−x)Ba2Cu3O7−δ(Pr_xY_{1-x})Ba_2Cu_3O_{7-\delta} : Influence of magnetic scattering

We report measurements of the surface impedance $Z_s=R_s+iX_s$ of $(Pr_xY_{1-x})Ba_2Cu_3O_{7-\delta}$, $(x=0,0.15,0.23,0.3,0.4,0.5)$. Increasing $Pr$ concentration leads to some striking results not observed in samples doped by non-magnetic constituents. The three principal features of the $R_s(T)$ data - multiple structure in the transition, a high residual resistance and, at high $Pr$ concentrations, an upturn of the low $T$ data, are all characteristic of the influence of magnetic scattering on superconductivity, and appear to be common to materials where magnetism and superconductivity coexist. The low $T$ behavior of $\lambda (T)$ appears to change from $T$ to $T^4$ at large $Pr$ doping, and provides evidence of the influence of magnetic pairbreaking of the $Pr$.Comment: 5 pages, 3 eps figures, Revtex, 2-column format, uses graphicx. To appear in Physica C. Postscript version also available at http://sagar.physics.neu.edu/preprints.htm

M-theory on `toric' G_2 cones and its type II reduction

We analyze a class of conical G_2 metrics admitting two commuting isometries, together with a certain one-parameter family of G_2 deformations which preserves these symmetries. Upon using recent results of Calderbank and Pedersen, we write down the explicit G_2 metric for the most general member of this family and extract the IIA reduction of M-theory on such backgrounds, as well as its type IIB dual. By studying the asymptotics of type II fields around the relevant loci, we confirm the interpretation of such backgrounds in terms of localized IIA 6-branes and delocalized IIB 5-branes. In particular, we find explicit, general expressions for the string coupling and R-R/NS-NS forms in the vicinity of these objects. Our solutions contain and generalize the field configurations relevant for certain models considered in recent work of Acharya and Witten.Comment: 45 pages, references adde

Self-similar stable processes arising from high-density limits of occupation times of particle systems

We extend results on time-rescaled occupation time fluctuation limits of the $(d,\alpha, \beta)$-branching particle system $(0<\alpha \leq 2, 0<\beta \leq 1)$ with Poisson initial condition. The earlier results in the homogeneous case (i.e., with Lebesgue initial intensity measure) were obtained for dimensions $d>\alpha / \beta$ only, since the particle system becomes locally extinct if $d\le \alpha / \beta$. In this paper we show that by introducing high density of the initial Poisson configuration, limits are obtained for all dimensions, and they coincide with the previous ones if $d>\alpha/\beta$. We also give high-density limits for the systems with finite intensity measures (without high density no limits exist in this case due to extinction); the results are different and harder to obtain due to the non-invariance of the measure for the particle motion. In both cases, i.e., Lebesgue and finite intensity measures, for low dimensions ($d<\alpha(1+\beta)/\beta$ and $d<\alpha(2+\beta)/(1+\beta)$, respectively) the limits are determined by non-L\'evy self-similar stable processes. For the corresponding high dimensions the limits are qualitatively different: ${\cal S}'(R^d)$-valued L\'evy processes in the Lebesgue case, stable processes constant in time on $(0,\infty)$ in the finite measure case. For high dimensions, the laws of all limit processes are expressed in terms of Riesz potentials. If $\beta=1$, the limits are Gaussian. Limits are also given for particle systems without branching, which yields in particular weighted fractional Brownian motions in low dimensions. The results are obtained in the setup of weak convergence of S'(R^d)$-valued processes.Comment: 28 page