Quartic monoid surfaces with maximum number of lines

In 1884 the German mathematician Karl Rohn published a substantial paper (Rohn, 1884) 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⊆PK1 (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 S3 except for one surface of the family, whose stabiliser is a group isomorphic to S3×C3. Finally, given two quartic surfaces Q(a) and Q(b), with a,b∈A, we show that Q(a) and Q(b) are projectively equivalent if and only if j(a)=j(b), where j is the j-function. To get our results, several computational tools, available in computer algebra systems, are used.