Related papers: Higher genus icosahedral Painleve curves
The gonality conjecture predicts that the gonality of a curve can be read off Koszul cohomology of line bundles of sufficiently large degree. We verify this conjecture for generic curves of odd genus. The even-genus case was previously…
We unconditionally determine $I_\Q(d)$, the set of possible prime degrees of cyclic $K$-isogneies of elliptic curves with $\Q$-rational $j$-invariants and without complex multiplication over number fields $K$ of degree $\leq d$, for $d\leq…
In an earlier article, we presented a method to obtain integrals of motion and polynomial algebras for a class of two-dimensional superintegrable systems from creation and annihilation operators. We discuss the general case and present its…
We study a relationship between two genus 2 curves whose jacobians are isogenous with kernel equal to a maximal isotropic subspace of p-torsion points with respect to the Weil pairing. For p = 3 we find an explicit relationship between the…
The Painleve-IV equation has two families of rational solutions generated respectively by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational…
A viable and still unproved conjecture states that, if $X$ is a smooth algebraic surface and $C$ is a smooth algebraic curve in $X$, then $C$ realizes the smallest possible genus amongst all smoothly embedded $2$-manifolds in its homology…
In a previous paper, we classified and constructed all rational plane curves of type (d,d-2). In this paper, we generalize these results to irreducible plane curves of type (d,d-2) with positive genus.
We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve $y^4 = x^5 + \lambda_4x^4 + \lambda_3x^3 + \lambda_2x^2 + \lambda_1x + \lambda_0$. We construct Abelian…
This paper is a continuation of our analysis, begun in arXiv:1310.2276, of the rational solutions of the inhomogeneous Painleve-II equation and associated rational solutions of the homogeneous coupled Painleve-II system in the limit of…
The moduli spaces of trigonal curves of odd genus $g>4$ are proven to be rational.
We classify the SU(2)-invariant anti-self-dual metrics with a signature (+,+,-,-). The metrics are specified by a solution of Painleve VI, V, III or II. Moreover we show the geometric meaning of the metrics specified by each type of…
The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus g_0 or g_1 have been characterized up to isomorphism. A natural genus to be studied…
In this paper we prove a general theorem concerning the number of translation classes of curves of genus $g$ belonging to a fixed cohomology class in a polarized abelian variety of dimension $g$. For $g = 2$ we recover results of G\"ottsche…
Given a smooth and projective curve C and a smooth and projective toric variety X, we first describe a compactification of the space of morphisms from C to X representing a fixed homology class, and after we study the intersection theory on…
We give a method to construct stable vector bundles whose rank divides the degree over curves of genus bigger than one. The method complements the one given by Newstead. Finally, we make some systematic remarks and observations in…
This thesis reviews the theory of bundle gerbes and then examines the higher dimensional notion of a bundle 2-gerbe. The notion of a bundle 2-gerbe connection and 2-curving are introduced and it is shown that there is a class in…
We study the global analytic properties of the solutions of a particular family of Painleve' VI equations with the parameters $\beta=\gamma=0$, $\delta={1\over2}$ and $\alpha$ arbitrary. We introduce a class of solutions having critical…
The $q$-Painlev\'e equation of type $E^{(1)}_6$ is obtained by Pad\'e method. Special solutions in determinant formula to the $q$-Painlev\'e equation is presented. A relation between Pad\'e method and B\"acklund transformation of type…
Given an ordered sequence of $N$-choose-2 integers, we give necessary and sufficient conditions to have an ordered collection of $N$ simple closed curves on a torus such that the algebraic pairwise intersections of those curves are the…
Suppose $\mathcal{X}$ is an $n$-correct set of nodes in the plane, that is, it admits a unisolvent interpolation with bivariate polynomials of total degree less than or equal to $n.$ Then an algebraic curve $q$ of degree $k\le n$ can pass…