The Curve of “prym Canonical” Gauss Divisors on a Prym Theta Divisor

Introduction: A good understanding of the geometry of a theta divisor Θ of a principally polarized abelian variety (A,Θ) requires a knowledge of properties of its canonical linear system, the Gauss linear system |OΘ(Θ)|. A striking feature of the theta divisor Θ(C) of the Jacobian of a curve C is that the dual of the branch divisor of the associated Gauss map γΘ on Θ, is not a hypersurface as expected but a non degenerate curve [A], namely the canonical model Cω ⊂ |ωC |∗ ∼= |OΘ(Θ)| of C. This feature is so striking that one is led to ask whether it is shared by other principally polarized abelian varieties, for example by those p.p.a.v.’s most similar to Jacobians, the Prym varieties. For the Prym variety (P,Ξ) of a connected étale double cover of curves π : C̃ → C the most natural first question of this type is whether the branch divisor of γΞ is dual to the Prym canonical model φη(C) = Cη ⊂ |ωC ⊗ η|∗ of the curve C. Specialization to a Jacobian (an approach proposed in private communication by Donagi), e.g. by letting C become trigonal, seems to imply that the set of Prym varieties whose Gauss map γΞ has branch divisor dual to Cη is a union of proper subvarieties of all Pryms, but gives no information on the possible number of such subvarieties. One obstacle to proving a more precise result has been a lack of computable models for the divisors in the Gauss linear system on a Prym theta divisor. In the present paper we construct explicit models for the Gauss divisors Γp parametrized by points φη(p) of the Prym canonical curve Cη and deduce from their geometry that in fact Cη is never dual to a component of the branch divisor of γΞ, for any connected étale double cover C̃ → C of any non hyperelliptic curve C of genus g ≥ 4. (These may be considered as limiting cases of the models Spq given in [BD2, proof of Prop. 1, p.615] for proper intersections of translates of theta divisors on Prym varieties.) Although this means one cannot repeat for any Prym varieties Andreotti’s proof [A] of the Torelli theorem, it raises the question of whether the property that the dual of the branch divisor of the Gauss map is a non degenerate curve may be characteristic of Jacobians. This question remains open even among Prym varieties. A result of Beauville and Debarre stated in [BD2, Remarque 1, p.619], along with a result in the present paper, seems to imply that the set of Prym varieties of dimension ≥ 4 whose Gauss map γΞ has branch divisor dual to any curve is a union of proper subvarieties of all Pryms, and that Jacobians are an irreducible component of this union, but again leaves open the possible number of such subvarieties. An alternate model for the Gauss map on a Prym theta divisor given by Verra in [Ve], although complicated, has enabled him to compute the degree of γΞ for a generic Prym, and could eventually be useful in determining the branch divisor. For generic four dimensional Pryms, i.e. all generic p.p.a.v.’s of dimension four, Adams, McCrory, Shifrin,