Projective normality and the generation of the ideal of an Enriques surface

TL;DR

This paper establishes criteria for when a smooth Enriques surface in projective space is defined by quadrics, explores the structure of these quadrics, and provides a new proof of the surface's projective normality for degrees at least 12.

Contribution

It offers necessary and sufficient conditions for Enriques surfaces to be intersections of quadrics and introduces a new proof of their projective normality.

Findings

Criteria for Enriques surfaces to be intersections of quadrics

Description of the union of S and 2-planes when containing cubic curves

A quick proof of projective normality for degree ≥ 12

Abstract

We give necessary and sufficient criteria for a smooth Enriques surface S in P^r to be scheme-theoretically an intersection of quadrics. Moreover we prove in many cases that, when S contains plane cubic curves, the intersection of the quadrics containing S is the union of S and the 2-planes spanned by the plane cubic curves. We also give a new (very quick) proof of the projective normality of S if the degree of S is at least 12.