## Journal of Differential Geometry

### 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.

#### Article information

Source
J. Differential Geom., Volume 88, Number 3 (2011), 483-518.

Dates
First available in Project Euclid: 15 November 2011

https://projecteuclid.org/euclid.jdg/1321366357

Digital Object Identifier
doi:10.4310/jdg/1321366357

Mathematical Reviews number (MathSciNet)
MR2844440

Zentralblatt MATH identifier
1238.14026

#### Citation

Knutsen, Andreas Leopold; Lopez, Angelo Felice; Muñoz, Roberto. On the Extendability of Projective Surfaces and a Genus Bound for Enriques-Fano Threefolds. J. Differential Geom. 88 (2011), no. 3, 483--518. doi:10.4310/jdg/1321366357. https://projecteuclid.org/euclid.jdg/1321366357