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

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.