Singularities of Normal Quartic Surfaces I (char= 2)

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.