Classification of normal quartic surfaces with irrational singularities

Abstract

If a normal quartic surface admits a singular point that is not a rational double point, then the surface is determined by the triplet $(M,D,E)$ consisting of the minimal desingularization $M$, the pullback $D$ of a general hyperplane section, and a nonzero effective anti-canonical divisor $E$ of $M$. Geometric constructions of all the possible triplets $(M,D,E)$ are given.