The number of singular points of quartic surfaces (char=2)

Abstract

We show 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 $20$, and that if equality is attained, then the minimal resolution of $X$ is a supersingular K3 surface and the singular points are $20$ nodes. We produce examples with 14 nodes. In a sequel to this paper (in two parts, the second in collaboration with Matthias Sch\"utt) we show that the optimal bound is indeed 14, and that if equality is attained, then the minimal resolution of $X$ is a supersingular K3 surface and the singular points are $14$ nodes. We also obtain some smaller upper bounds under several geometric assumptions holding at one of the singular points $P$ (structure of tangent cone, separability/inseparability of the projection with centre $P$).