Some Remarks on Prym-Tyurin Varieties

Parigi, Giuliano

Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 1055-1069

Résumé

The aims of the present paper can be described as follows: a) In [2] Beauville showed that if some endomorphism $u$ a Jacobian $J(C)$ has connected kernel, the principal polarization on $J(C)$ induces a multiple of the principal polarization on the image of $u$. We reformulate and complete this theorem proving "constructively" the following: Theorem. Let $Z \subset J(C)$ be an abelian subvariety and $Y$ its complementary variety. $Z$ is a Prym-Tyurin variety with respect to $J(C)$ if and only if the following sequence $0 \to Y \hookrightarrow J(C) \to Z \to 0$ is exact. b) In [5] Izadi set the question whether every p.p.a.v. is a Prym-Tyurin variety for a symmetric fixed point free correspondence. In this work a contribution to a possible negative answer to this question is provided by building a classical Prym-Tyurin variety explicitly, but this variety can never be defined through a fixed point free correspondence.