Singularities of normal quartic surfaces I (char=2)

TL;DR

This paper establishes an upper bound of 16 on the number of singular points of normal quartic surfaces in characteristic 2, providing examples with 12 and 14 singularities and exploring geometric conditions that reduce this maximum.

Contribution

It determines the maximal number of singular points on such surfaces in characteristic 2 and analyzes how geometric assumptions influence this bound.

Findings

Maximum of 16 singular points for normal quartic surfaces in characteristic 2.

Existence of examples with 14 and 12 singular points.

Geometric conditions can lower the upper bound on singularities.

Abstract

We show, in this first 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 $16$. We produce examples with $14$, respectively $12$, singular points and show that, under several geometric assumptions ($\mathfrak S_4$-symmetry, or behaviour of the Gauss map, or structure of tangent cone at one of the singular points $P$, separability/inseparability of the projection with centre $P$), we can obtain smaller upper bounds for the number of singular points of $X$.