Enriques surfaces of non-degeneracy 3

Abstract

We classify all non-extendable 3-sequences of half-fibers on Enriques surfaces. If the characteristic is different from 2, we prove in particular that every Enriques surface admits a 4-sequence, which implies that every Enriques surface is the minimal desingularization of an Enriques sextic, and that every Enriques surface is birational to a Castelnuovo quintic. CONTENTS1. Introduction 1 2. c-sequences 4 3. Special 3-sequences 6 4. Non-extendable 3-sequences 13 4.1. Examples 13 4.2. Special non-extendable 3-sequences 17 4.3. Non-special non-extendable 3-sequences 19 5. Projective models of Enriques surfaces 22 References 23

On extra-special Enriques surfaces

“…Extending 3-sequences to 4-sequences is a much more difficult problem, even in characteristic p = 2. The classification of non-extendable 3-sequences is the subject of our follow-up article [11].…”

“…The three pencils of residual conics that we obtain from the latter three lines give rise to the genus one fibrations |2F 1 |, |2F 2 |, and |2F 3 |, and the preimages of the former six lines are the six half-fibers. We refer the reader to our follow-up article [11] for a closer study of the so-called special 3-sequences that give rise to this geometric situation.…”