Quartic monoid surfaces with maximum number of lines. (March 2022) - Beltrametti, Mauro Carlo; Logar, Alessandro; Torrente, Maria Laura
Abstract: 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 ⊆ P K 1 ( 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, 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.