Plane curves with a large linear automorphism group in characteristic $p$

TL;DR

This paper classifies plane curves invariant under large subgroups of PGL(3,q), analyzing their degrees, geometric features, and automorphism groups, revealing new insights into their structure in characteristic p.

Contribution

It provides a detailed classification of G-invariant curves for maximal subgroups of PGL(3,q), including degree bounds, geometric properties, and automorphism group structures.

Findings

Curves of minimal degree form a pencil depending on G

Invariant curves exhibit Frobenius nonclassicality

Examples of nonlinear automorphism groups are identified

Abstract

Let $G$ be a subgroup of the three dimensional projective group $\mathrm{PGL}(3,q)$ defined over a finite field $\mathbb{F}_q$ of order $q$, viewed as a subgroup of $\mathrm{PGL}(3,K)$ where $K$ is an algebraic closure of $\mathbb{F}_q$. For the seven nonsporadic, maximal subgroups $G$ of $\mathrm{PGL}(3,q)$, we investigate the (projective, irreducible) plane curves defined over $K$ that are left invariant by $G$. For each, we compute the minimum degree $d(G)$ of $G$-invariant curves, provide a classification of all $G$-invariant curves of degree $d(G)$, and determine the first gap $\varepsilon(G)$ in the spectrum of the degrees of all $G$-invariant curves. We show that the curves of degree $d(G)$ belong to a pencil depending on $G$, unless they are uniquely determined by $G$. We also point out that $G$-invariant curves of degree $d(G)$ have particular geometric features such as Frobenius nonclassicality and an unusual variation of the number of $\mathbb{F}_{q^i}$-rational points. For most examples of plane curves left invariant by a large subgroup of $\mathrm{PGL}(3,q)$, the whole automorphism group of the curve is linear, i.e., a subgroup of $\mathrm{PGL}(3,K)$. Although this appears to be a general behavior, we show that the opposite case can also occur for some irreducible plane curves, that is, the curve has a large group of linear automorphisms, but its full automorphism group is nonlinear.