Prym map and second gaussian map for Prym-canonical line bundles

TL;DR

This paper investigates the Prym map and the second Gaussian map for Prym-canonical bundles, proving surjectivity for general cases in high genus through degeneration techniques.

Contribution

It establishes a connection between the Prym map's second fundamental form and the second Gaussian map, and proves surjectivity for high genus cases.

Findings

Second Gaussian map is surjective for general [C,A] in R_g for g > 19.

The second fundamental form of the Prym map lifts the second Gaussian map.

Degeneration to binary curves is used to prove surjectivity.

Abstract

We show that the second fundamental form of the Prym map lifts the second gaussian map of the Prym-canonical bundle. We prove, by degeneration to binary curves, that this gaussian map is surjective for the general point [C,A] of R_g for g > 19.