Related papers: Rank 2 integrable systems of Prym varieties

A new relation between Prym varieties of arbitrary morphisms of algebraic curves and integrable systems is discovered. The action of maximal commutative subalgebras of the formal loop algebra of GL(n) defined on certain infinite-dimensional…

A Hamiltonian system is completely integrable (in the sense of Liouville) if there exist as many independent integrals of motion in involution as the dimension of the configuration space. Under certain regularity conditions,…

Three new examples of 4-dimensional irreducible symplectic V-manifolds are constructed. Two of them are relative compactified Prymians of a family of genus-3 curves with involution, and the third one is obtained from a Prymian by Mukai's…

Let X be a smooth complex surface of general type such that the image of the canonical map $\phi$ of X is a surface $\Sigma$ and that $\phi$ has degree $\delta\geq 2$. Let $\epsilon\colon S\to \Sigma$ be a desingularization of $\Sigma$ and…

In this work we study quantitative existence results for genus-$2$ curves over $\mathbb{Q}$ whose Jacobians have Mordell-Weil rank at least $1$ or $2$, ordering the curves by the naive height of their integral Weierstrass models. We use…

In recent work, we presented the construction of a family of difference equations associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus $g$. As well as proving that each such…

We construct a three-parameter family of non-hyperelliptic and bielliptic plane genus-three curves whose associated Prym variety is two-isogenous to the Jacobian variety of a general hyperelliptic genus-two curve. Our construction is based…

Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let $\mathbb L$ be a rank $2$, geometrically irreducible $\bar{\mathbb Q}_\ell$-local system on $U$ with cyclotomic determinant that extends to an…

We study unramified Galois $\mathbb{Z}_3 \times \mathbb{Z}_3$ coverings of genus 2 curves and the corresponding Prym varieties and Prym maps. In particular, we prove that any such covering can be reconstructed from its Prym variety, that…

Recently Gubbiotti, Joshi, Tran and Viallet classified birational maps in four dimensions admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. The…

Associated to a convex integral polygon $N$ in the plane are two integrable systems: the cluster integrable system of Goncharov and Kenyon, constructed from the dimer model on bipartite torus graphs, and the Beauville integrable system…

Bi-presymplectic chains of one-forms of arbitrary co-rank are considered. The conditions in which such chains represent some Liouville integrable systems and the conditions in which there exist related bi-Hamiltonian chains of vector fields…

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…

We consider an integrable system in five unknowns having three quartics invariants. We show that the complex affine variety defined by putting these invariants equal to generic constants, completes into an abelian surface; the jacobian of a…

We consider generalisations of the elliptic Calogero--Moser systems associated to complex crystallographic groups in accordance to [1]. In our previous work [2], we proposed these systems as candidates for Seiberg--Witten integrable systems…

We discuss the geometry of the genus one fibrations associated to an elliptic fibration on a K3 surface. We show that the two-torsion subgroup of the Brauer group of a general elliptic fibration is naturally isomorphic to the two-torsion of…

In this paper we consider the Prym variety $P(\widetilde{C}/C)$ associated to a Galois coverings of curves $f:\widetilde{C}\to C$ branched at $r$ points. We discuss some properties and equivalent definitions and then consider the Prym map…

We introduce two algebraic completely integrable analogues of the Mumford systems which we call hyperelliptic Prym systems, because every hyperelliptic Prym variety appears as a fiber of their momentum map. As an application we show that…

We study deformations of irreducible Hermitian symmetric spaces $S$ of the compact type, known to be locally rigid, as projective-algberaic manifolds and prove that no jump of complex structures can occur. For each $S$ of rank $\ge 2$ there…

The modular variety of non singular and complete hyperelliptic curves with level-two structure of genus 3 is a 5-dimensional quasi projective variety which admits several standard compactifications. The first one, X, comes from the…