On the first Gaussian map for Prym-canonical line bundles

TL;DR

This paper investigates the properties of the first Gaussian map for Prym-canonical line bundles, establishing conditions for its surjectivity and injectivity depending on the genus of the curve.

Contribution

It proves the surjectivity of the first Gaussian map for general Prym-canonical line bundles when genus exceeds 11, and injectivity when genus is less than 12.

Findings

Surjective Gaussian map for g > 11

Injective Gaussian map for g < 12

Uses degeneration to Prym-canonical binary curves

Abstract

We prove by degeneration to Prym-canonical binary curves that the first Gaussian map of the Prym canonical line bundle $\omega_C \otimes A$ is surjective for the general point [C,A] of R_g if g >11, while it is injective if g < 12.