Related papers: Generic hyperelliptic Prym varieties in a generali…

For a class of non-hyperelliptic genus 3 curves C which are 2-fold coverings of elliptic curves E, we give an explicit algebraic description of all birationally non-equivalent genus 2 curves whose Jacobians are degree 2 isogeneous to the…

We study Prym varieties of ramified (at precisely two points) double covers of smooth irreducible complex projectives curves that admit an automorphism of prime order $p>2$. Using Galois theory, we give an explicit constructions of Prym…

Consider an ordinary differential equation which has a Lax pair representation A'(x)= [A(x),B(x)], where A(x) is a matrix polynomial with a fixed regular leading coefficient and the matrix B(x) depends only onA(x). Such an equation can be…

We describe the isotypical decomposition of the Jacobian variety JW of the Galois extension W-->T of any fourfold cover of smooth connected irreducible projective complex curves X-->T, in terms of Prym's of intermediate covers. We also…

We study an explicit $(2g-1)$-dimensional family of Jacobian varieties of dimension $\frac{d-1}2(g-1)$, arising from quotient curves of unramified cyclic coverings of prime degree $d$ of hyperelliptic curves of genus $g\ge 2$. By using a…

It is well known that the Prym variety of an \'etale cyclic covering of a hyperelliptic curve is isogenous to the product of two Jacobians. Moreover, if the degree of the covering is odd or congruent to 2 mod 4, then the canonical isogeny…

Let $\phi:\,X\rightarrow Y$ be a (possibly ramified) cover between two algebraic curves of positive genus. We develop tools that may identify the Prym variety of $\phi$, up to isogeny, as the Jacobian of a quotient curve $C$ in the Galois…

We develop a cohomological description of various explicit descents in terms of generalized Jacobians, generalizing the known description for hyperelliptic curves. Specifically, given an integer $n$ dividing the degree of some reduced…

The Prym map assigns to each covering of curves a polarized abelian variety. In the case of unramified cyclic covers of curves of genus two, we show that the Prym map is ramified precisely on the locus of bielliptic covers. The key…

In this survey of works on a characterization of Jacobians and Prym varieties among indecomposable principally polarized abelian varieties via the soliton theory we focus on a certain circle of ideas and methods which show that the…

The classical definition of Prym varieties deals with the unramified covers of curves. The aim of the present paper is to give explicit algebraic descriptions of the Prym varieties associated to ramified double covers of algebraic curves.…

Let $\pi: Z \ra X$ be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply-connected Lie group $G$. For any dominant weight $\lambda$ consider the curve $Y = Z/\Stab(\lambda)$. The Kanev…

We study geometric and algebraic geometric properties of the continuous and discrete Neumann systems on cotangent bundles of Stiefel varieties $V_{n,r}$. The systems are integrable in the non-commutative sense, and by applying a $2r\times…

We investigate the number and the geometry of smooth hyperelliptic curves on a general complex abelian surface. We show that the only possibilities of genera of such curves are $2,3,4$ and $5$. We focus on the genus 5 case. We prove that up…

Many finite dimensional integrable systems qre expressed with the help of the Lax equation which highlights a spectral parameter and therefore a spectral curve. These spectral curves are the starting point of an algebro-geometric…

Given an Enriques surface $T$, its universal K3 cover $f: S\to T$, and a genus $g$ linear system $|C|$ on $T$, we construct the relative Prym variety $P_H=\Prym_{v, H}(\D/\CC)$, where $\CC\to |C|$ and $\D\to |f^*C|$ are the universal…

Let $G$ denote a finite group and $\pi: Z \to Y$ a Galois covering of smooth projective curves with Galois group $G$. For every subgroup $H$ of $G$ there is a canonical action of the corresponding Hecke algebra $\mathbb{Q}[H \backslash…

It is well-known that abelian varieties are projective, and so that there exist explicit polynomial and rational functions which define both the variety and its group law. It is however difficult to find any explicit polynomial and rational…

In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This…

The Prym variety for a branched double covering of a nonsingular projective curve is defined as a polarized abelian variety. We prove that any double covering of an elliptic curve which has more than $4$ branch points is recovered from its…