On the Extendability of Projective Surfaces and a Genus Bound for Enriques-Fano Threefolds

Abstract

We introduce a technique based on Gaussian maps to study whether a surface can lie on a threefold as a very ample divisor with given normal bundle. We give applications, among which one to surfaces of general type and another to Enriques surfaces. In particular, we prove the genus bound $g \le17$ for Enriques-Fano threefolds. Moreover we find a new Enriques-Fano threefold of genus 9 whose normalization has canonical but not terminal singularities and does not admit $\mathbb{Q}$-smoothings.