Plane curves with a large linear automorphism group in characteristic p

Abstract

Let G be a subgroup of the three dimensional projective group $\text{PGL}(3,q)$ defined over a finite field $F_q$ of order q, viewed as a subgroup of $\text{PGL}(3,K)$ where K is an algebraic closure of $F_q$. For $G\cong \text{PGL}(3,q)$ and for the seven nonsporadic, maximal subgroups G of $\text{PGL}(3,q)$, we investigate the (projective, irreducible) plane curves defined over K that are left invariant by G. For each, we compute the minimum degree $d\left(G\right)$ of G-invariant curves, provide a classification of all G-invariant curves of degree $d\left(G\right)$, and determine the first gap $\epsilon \left(G\right)$ in the spectrum of the degrees of all G-invariant curves. We show that the curves of degree $d\left(G\right)$ belong to a pencil depending on G, unless they are uniquely determined by G. For most examples of plane curves left invariant by a large subgroup of $\text{PGL}(3,q)$, the whole automorphism group of the curve is linear, i.e., a subgroup of $\text{PGL}(3,K)$. 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.