Quartic monoid surfaces with maximum number of lines

Abstract

In 1884 the German mathematician Karl Rohn published a substantial paper on [11] on the properties of quartic surfaces with triple points, proving (among many other things) that the maximum number of lines contained in a quartic monoid surface is 31.In this paper we study in details this class of surfaces. We prove that there exists an open subset A ⊆ P 1 K (K is a characteristic zero field) that parametrizes (up to a projectivity) all the quartic monoid surfaces with 31 lines; then we study the action of PGL(4, K) on these surfaces, we show that the stabiliser of each of them is a group isomorphic to S 3 except for one surface of the family, whose stabiliser is a group isomorphic to S 3 × C 3 . Finally we show that the j-invariant allows one to decide, also in this situation, when two elements of A give the same surface up to a projectivity.To get our results, several computational tools, available in computer algebra systems, are used.