On the Hodge cycles of Prym varieties

Abstract

We show that the Neron–Severi group of the Prym variety for a degree three unramified Galois covering of a hyperelliptic Riemann surface has a distinguished subgroup of rank three. For the general hyperelliptic curve, the algebra of Hodge cycles on the Prym variety is generated by this group of rank three. Mathematics Subject Classification (2000): 14H40 (primary); 14C30 (secondary). 1. – Introduction Let X be a hyperelliptic Riemann surface. Let f : Y −→ X be a unramified Galois covering with Galois group Z/3Z with Y connected. The Neron–Severi group of the Prym variety Prym( f ) has a certain distinguished subgroup of rank three. This subgroup is constructed as follows. We first prove that Prym( f ) is identified with Pic0(Z) × Pic0(Z), where Z is a Riemann surface constructed from the covering datum (Lemma 3.1). The two projections and the addition law of Pic0(Z) together give three maps from Pic0(Z)×Pic0(Z) to Pic0(Z). The pull backs, by these three maps, to Pic0(Z)×Pic0(Z) = Prym( f ) of the natural polarization on Pic0(Z) give the distinguished subgroup of the Neron–Severi group of Prym( f ). For the general hyperelliptic curve X , the algebra of Hodge cycles on Prym( f ) is generated by this subgroup of rank three of NS(Prym( f )) (Theorem 4.1). In particular, for general hyperelliptic X , the Neron–Severi group NS(Prym( f )) is of rank three, and all the Hodge cycles on Prym( f ) are algebraic. We also consider degree three ramified coverings of an arbitrary compact Riemann surface. Under an assumption on the covering, the algebra of Hodge cycles of the Prym variety for the general Riemann surface is again generated by the Neron–Severi group (Lemma 5.1). Pervenuto alla Redazione il 9 luglio 2003 e in forma definitiva il 25 giugno 2004.