Products of Jacobians as Prym-Tyurin varieties

TL;DR

This paper demonstrates that the product of Jacobian varieties of certain algebraic curves can be endowed with a Prym-Tyurin structure of explicitly computable smaller exponent, providing new insights into their geometric and algebraic properties.

Contribution

It constructs explicit Prym-Tyurin structures of smaller exponent on products of Jacobians of curves, improving known bounds and offering explicit correspondences.

Findings

Product of Jacobians admits Prym-Tyurin structure of exponent n^{m-1}

Exponent is smaller than previously known for arbitrary abelian varieties

Explicit correspondences are provided for the Prym-Tyurin structures

Abstract

Let $X_1, ..., X_m$ denote smooth projective curves of genus $g_i \geq 2$ over an algebraically closed field of characteristic 0 and let $n$ denote any integer at least equal to $1+\max_{i=1}^m g_i$. We show that the product $JX_1 \times ... \times JX_m$ of the corresponding Jacobian varieties admits the structure of a Prym-Tyurin variety of exponent $n^{m-1}$. This exponent is considerably smaller than the exponent of the structure of a Prym-Tyurin variety known to exist for an arbitrary principally polarized abelian variety. Moreover it is given by explicit correspondences.