Quartic monoid surfaces with maximum number of lines. (March 2022)

• Record Type: Journal Article
• Authors: Beltrametti, Mauro Carlo; Logar, Alessandro; Torrente, Maria Laura
• Is Part Of: Journal of symbolic computation. Volume 109(2022)
• Page Start: 250
• Page End: 258
• Publication Date: 2022-03
• Subjects: Quartic monoid surface -- Algebraic surface -- Incidence structure -- Constructive geometry -- j-function; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2020.07.005
• Languages: English
• ISSNs: 0747-7171

Abstract

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.