Gauss-Prym maps on Enriques surfaces

TL;DR

This paper proves the surjectivity of the $k$-th Gaussian map on certain Enriques surfaces and their hyperplane sections, extending understanding of Gaussian maps in algebraic geometry.

Contribution

It establishes new conditions for the surjectivity of Gaussian maps on unnodal Enriques surfaces and their hyperplane sections, advancing the theory of Gaussian maps.

Findings

Surjectivity of the $k$-th Gaussian map under $(H)>2k+4$

Surjectivity of the $k$-th Gauss-Prym map when $(H)>4(k+2)$

Special case for $k=1$ with $(H)>6$

Abstract

We prove that the $k$-th Gaussian map $\gamma^k_{H}$ is surjective on a polarized unnodal Enriques surface $(S, H)$ with $\phi(H)>2k+4$. In particular, as a consequence, when $\phi(H)>4(k+2)$, we obtain the surjectivity of the $k$-th Gauss-Prym map $\gamma^k_{\omega_C\otimes\alpha}$ on smooth hyperplane sections $C\in \vert H\vert.$ In case $k=1$ it is sufficient to ask $\phi(H)>6$.