Singularities of normal quartic surfaces II (char=2)

Fabrizio Catanese (Lehrstuhl Mathematik VIII, Mathematisches Institut der Universität, Bayreuth, Germany; and Korea Institute for Advanced Study, Seoul, South Korea)

Matthias Schütt (Institut für Algebraische Geometrie, Leibniz Universität Hannover, Germany; and Riemann Center for Geometry and Physics, Leibniz Universität Hannover, Germany)

Abstract

We show, in this second part, that the maximal number of singular points of a normal quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic $2$ is at most $14$, and that, if we have $14$ singularities, these are nodes and moreover the minimal resolution of $X$ is a supersingular K3 surface.

We produce an irreducible component, of dimension $24$, of the variety of quartics with $14$ nodes. We also exhibit easy examples of quartics with $7$ $A_3$-singularities.

Keywords

quartic surface, singularity, Gauss map, genus one fibration, supersingular K3 surface

2010 Mathematics Subject Classification

Primary 14J17, 14J28. Secondary 14J25, 14N05.

Full Text (PDF format)

The first author acknowledges support of the ERC 2013 Advanced Research Grant – 340258 – TADMICAMT.

Received 6 October 2021

Received revised 3 March 2022

Accepted 23 May 2022

Published 25 October 2022