Transactions of the American Mathematical Society

Abstract:

The Gauss linear system on the theta divisor of the Jacobian of a nonhyperelliptic curve has two striking properties: 1) the branch divisor of the Gauss map on the theta divisor is dual to the canonical model of the curve; 2) those divisors in the Gauss system parametrized by the canonical curve are reducible. In contrast, Beauville and Debarre prove on a general Prym theta divisor of dimension $\ge 3$ all Gauss divisors are irreducible and normal. One is led to ask whether properties 1) and 2) may characterize the Gauss system of the theta divisor of a Jacobian. Since for a Prym theta divisor, the most distinguished curve in the Gauss system is the Prym canonical curve, the natural analog of the canonical curve for a Jacobian, in the present paper we analyze whether the analogs of properties 1) or 2) can ever hold for the Prym canonical curve. We note that both those properties would imply that the general Prym canonical Gauss divisor would be nonnormal. Then we find an explicit geometric model for the Prym canonical Gauss divisors and prove the following results using Beauville’s singularities criterion for special subvarieties of Prym varieties:

Theorem. For all smooth doubly covered nonhyperelliptic curves of genus $g\ge 5$, the general Prym canonical Gauss divisor is normal and irreducible.

Corollary. For all smooth doubly covered nonhyperelliptic curves of genus $g\ge 4$, the Prym canonical curve is not dual to the branch divisor of the Gauss map.