On the $PGL_2(q)$-orbits of lines of $PG(3,q)$ and binary quartic forms
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| Format: | Preprint |
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2023
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| _version_ | 1866909729708572672 |
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| author | Kaipa, Krishna Patanker, Nupur Pradhan, Puspendu |
| author_facet | Kaipa, Krishna Patanker, Nupur Pradhan, Puspendu |
| contents | 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 $\mathbb P^3$. In the Klein representation of lines of $\mathbb P^3$, the tangent lines of $C$ are represented by a degree $4$ rational normal curve in a hyperplane $\mathcal H$ of the second exterior power $\mathbb P^5$ of $\mathbb P^3$. Atiyah studied the lines of $\mathbb P^3$ with respect to $C$, in terms of the geometries of these two curves. Polar duality of lines on $\mathbb P^3$ corresponds to Hodge duality on $\mathbb P^5$, and $\mathcal H$ is the hyperplane of Hodge self-dual elements of $\mathbb P^5$. We show that $\mathcal H$ can be identified in a $PGL_2$-equivariant way with the space of binary quartic forms over $F$, and that pairs of polar dual lines of $\mathbb P^3$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_07118 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the $PGL_2(q)$-orbits of lines of $PG(3,q)$ and binary quartic forms Kaipa, Krishna Patanker, Nupur Pradhan, Puspendu Combinatorics Algebraic Geometry 51E20, 51N35, 14N10, 05B25, 05E10, 05E14, 05E18 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 $\mathbb P^3$. In the Klein representation of lines of $\mathbb P^3$, the tangent lines of $C$ are represented by a degree $4$ rational normal curve in a hyperplane $\mathcal H$ of the second exterior power $\mathbb P^5$ of $\mathbb P^3$. Atiyah studied the lines of $\mathbb P^3$ with respect to $C$, in terms of the geometries of these two curves. Polar duality of lines on $\mathbb P^3$ corresponds to Hodge duality on $\mathbb P^5$, and $\mathcal H$ is the hyperplane of Hodge self-dual elements of $\mathbb P^5$. We show that $\mathcal H$ can be identified in a $PGL_2$-equivariant way with the space of binary quartic forms over $F$, and that pairs of polar dual lines of $\mathbb P^3$ 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. |
| title | On the $PGL_2(q)$-orbits of lines of $PG(3,q)$ and binary quartic forms |
| topic | Combinatorics Algebraic Geometry 51E20, 51N35, 14N10, 05B25, 05E10, 05E14, 05E18 |
| url | https://arxiv.org/abs/2312.07118 |