Polarization types of isogenous Prym-Tyurin varieties

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.