Singularities of normal quartic surfaces II (char=2)

TL;DR

This paper establishes that quartic surfaces over algebraically closed fields of characteristic 2 can have at most 14 singular points, which are nodes, and explores the geometric structure of such surfaces, including their resolutions.

Contribution

It proves the maximum number of singularities on quartic surfaces in characteristic 2 and characterizes the nature of these singularities, including the structure of the minimal resolution.

Findings

Maximum 14 singular points on quartic surfaces in characteristic 2

Surfaces with 14 singularities are supersingular K3 surfaces

Constructed examples with 7 A3-singularities

Abstract

We show, in this second part, that the maximal number of singular points of a 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.