2023/12/12 by Krishna Kaipa, Kaipa, Krishna, Nupur Patanker +3 · 2 citations
Mathematics · #05B25 #05E10 #05E14 #05E18 #14N10 #51E20 #51N35 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2312.07118
openalex publication_date 2023/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the problem of classifying the lines of the projective 3-space PG(3,q) over a finite field GF(q) into orbits of the group G=PGL(2,q) of linear symmetries of the twisted cubic C. A generic line neither intersects C nor lies in any of its osculating planes. While the non-generic lines have been classified into G-orbits in literature, it has been an open problem to classify the generic lines into G-orbits. For a general field F of characteristic different from 2 and 3, the twisted cubic determines a symplectic polarity on \mathbb P3. In the Klein representation of lines of \mathbb P3, the tangent lines of C are represented by a degree 4 rational normal curve in a hyperplane \mathcal H of the second exterior power \mathbb P5 of \mathbb P3. Atiyah studied the lines of \mathbb P3 with respect to C, in terms of the geometries of these two curves. Polar duality of lines on \mathbb P3 corresponds to Hodge duality on \mathbb P5, and \mathcal H is the hyperplane of Hodge self-dual elements of \mathbb P5. We show that \mathcal H can be identified in a PGL2-equivariant way with the space of binary quartic forms over F, and that pairs of polar dual lines of \mathbb P3 correspond to binary quartic forms whose apolar invariant is a square. We first solve the open problem of classifying binary quartic forms over GF(q) into G-orbits, and then use it to solve the main problem.