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The Automorphism Group of Plane Algebraic Curves with Singer Automorphisms

Authors:	A Cossidente A Siciliano

Institution:	(1) Dipartimento di Matematica, Università della Basilicata, Contrada Macchia Romana, 85100 Potenza, Italy

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 X^q+1Y+Y^q+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 $PGL(3,\mathbb{F}_{q})$ .

Keywords:	algebraic curve singer cyclic group automorphisms
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