Algebraic curves with automorphism groups of large prime order

Abstract

Let ${\mathcal {X}}$ be a (projective, algebraic, non-singular, absolutely irreducible) curve of genus g defined over an algebraically closed field K of characteristic $p \ge 0$, and let q be a prime dividing the cardinality of $\text{ Aut }({\mathcal {X}})$. We say that ${\mathcal {X}}$ is a q-curve. Homma proved that either $q \le g+1$ or $q = 2g+1$, and classified $(2g+1)$-curves up to birational equivalence. In this note, we give the analogous classification for $(g+1)$-curves, including a characterization of hyperelliptic $(g+1)$-curves. Also, we provide the characterization of the full automorphism groups of q-curves for $q= 2g+1, g+1$. Here, we make use of two different techniques: the former case is handled via a result by Vdovin bounding the size of abelian subgroups of finite simple groups, the second via the classification by Giulietti and Korchmáros of automorphism groups of curves of even genus. Finally, we give some partial results on the classification of q-curves for $q = g, g-1$.