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

Abstract

Let G be a subgroup of the three dimensional projective group PGL(3, q) defined over a finite field Fq of order q, viewed as a subgroup of PGL(3, K) where K is an algebraic closure of Fq. For the seven nonsporadic, maximal subgroups G of 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 Ginvariant curves, provide a classification of all G-invariant curves of degree d(G), and determine the first gap ε(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 F q i -rational points. For most examples of plane curves left invariant by a large subgroup of PGL(3, q), the whole automorphism group of the curve is linear, i.e., a subgroup of 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.