Plane curves with a large linear automorphism group in characteristic p

Let G be a subgroup of the three dimensional projective group PGL(3 , q ) defined over a finite field F q of order q , viewed as a subgroup of PGL(3 , K ) where K is an algebraic closure of F q . For G similar to=PGL(3 , q ) and 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 G -invariant curves, provide a classification of all G -invariant curves of degree d ( G ), and determine the first gap epsilon ( 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 . 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. (c) 2024 Elsevier Inc. All rights reserved.