The Automorphism Group of Plane Algebraic Curves with Singer Automorphisms

Abstract

The main result in Cossidente and Siciliano (J. Number Theory, Vol. 99 (2003) pp. 373---382) states that if a Singer subgroup of PGL(3,q) is an automorphism group of a projective, geometric irreducible, non-singular plane algebraic curve $$\mathcal{X}$$ then either $$\deg(\mathcal{X})=q+2$$ or $$\deg(\mathcal {X})\ge q^2+q+1$$ . In the former case $$\mathcal{X}$$ is projectively equivalent to the curve $$\mathcal{X}_q$$ with equation Xq+1Y+Yq+1+X=0 studied by Pellikaan. Furthermore, the curve $$\mathcal{X}_q$$ has a very nice property from Finite Geometry point of view: apart from the three distinguished points fixed by the Singer subgroup, the set of its $$\mathbb{F}_{{q}^{3}}$$ -rational points can be partitioned into finite projective planes $$P^{2}(\mathbb{F}_{q})$$ . In this paper, the full automorphism group of such curves is determined. It turns out that $$Aut(\mathcal {X}_q)$$ is the normalizer of a Singer group in $$PGL(3,\mathbb{F}_{q})$$ .