Polarization types of isogenous Prym-Tyurin varieties

TL;DR

This paper investigates the polarization types of Prym-Tyurin varieties associated with certain curve coverings, providing explicit calculations for degree n=3 and conjecturing duality properties for specific cases.

Contribution

The paper computes polarization types of Prym-Tyurin varieties for degree n=3 and proves duality conjectures for étale coverings, extending known results from n=2.

Findings

Polarization type calculated for n=3 case.

Conjecture that P(X,δ) is dual to P(C,C') when Y=P^1.

Proved duality for n=3 and étale coverings.

Abstract

Let p:C-->Y be a covering of smooth, projective curves which is a composition of \pi:C-->C' of degree 2 and g:C'-->Y of degree n. Let f:X-->Y be the covering of degree 2^n, where the curve X parametrizes the liftings in C^{(n)} of the fibers of g:C'-->Y. Let P(X,\delta) be the associated Prym-Tyurin variety, known to be isogenous to the Prym variety P(C,C'). Most of the results in the paper focus on calculating the polarization type of the restriction of the canonical polarization of JX on P(X,\delta). We obtain the polarization type when n=3. When Y=P^1 we conjecture that P(X,\delta) is isomorphic to the dual of the Prym variety P(C,C'). This was known when n=2, we prove it when n=3, and for arbitrary n if \pi:C-->C' is \'{e}tale. Similar results are obtained for some other types of coverings.