2025/09/18 by Kaipa, Krishna, Pradhan, Puspendu
#05B25 #05E10 #05E14 #05E18 #14N10 #51E20 #51N35 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2509.15332
We consider the orbits of the group G=PGL2(q) on the points, lines and planes of the projective space PG(3,q) over a finite field \mathbb Fq of characteristic different from 2 and 3. The points of PG(3,q) can be identified with projective space of binary cubic forms, and the set \mathcal L of lines of PG(3,q) can be thought of as pencils of cubic forms. The action of G on PG(1,q) naturally induces an action of G on binary cubic forms f(X,Y). The points of PG(3, q) decompose into five G orbits. The G orbits on \mathcal L were recently obtained by the authors. Let \mathcal I be the subset of \mathcal L × PG(3,q) consisting of pairs (L,P) where L is a line incident with the point P. The decomposition of \mathcal L × PG(3,q) into G × G orbits yields a partition of \mathcal I. The problem that we solve in this work is to determine the sizes of the corresponding parts of \mathcal I.