The Brill-Noether curve and Prym-Tyurin varieties

TL;DR

This paper demonstrates that for a general curve of odd genus greater than 4, its Jacobian can be realized as a Prym-Tyurin variety associated with the Brill-Noether curve, enabling computation of certain divisor classes.

Contribution

It establishes a new geometric realization of Jacobians as Prym-Tyurin varieties linked to Brill-Noether curves for specific genera.

Findings

Jacobian of a general curve of genus 2a+1 is a Prym-Tyurin variety.

Computed the class of the sum of secant divisors for the curve.

Connected Prym-Tyurin varieties with Brill-Noether theory.

Abstract

We prove that the Jacobian of a general curve C of genus g=2a+1, with g>4, can be realized as a Prym-Tyurin variety for the Brill-Noether curve W^1_{a+2}(C). As a consequence of this result we are able to compute the class of the sum of the secant divisors for the curve C, embedded with a complete linear series g^{a-1}_{3a-2}