Plane non-singular curves with an element of "large" order in its automorphism group

TL;DR

This paper develops an algorithm to identify possible automorphism group orders for plane non-singular curves of fixed degree, and investigates the structure of these groups when the automorphism contains elements of 'large' or 'very large' order.

Contribution

It provides a method to determine which cyclic group orders can occur as automorphisms of plane curves of a given degree, and analyzes the automorphism groups with elements of large order.

Findings

Identifies divisibility conditions for automorphism group orders based on degree.

Classifies automorphism groups with elements of very large order.

Describes the loci of curves with specified automorphism group structures.

Abstract

In this note we determine, for an arbitrary but a fixed degree $d$, an algorithm to list the possible values $m$ for which $M_g^{Pl}(\mathbb{Z}/m)$ is non-empty where $\mathbb{Z}/m$ denotes the cyclic group of order $m$. In particular, we prove that $m$ should divide one of the integers: $d-1$, $d$, $d^2-3d+3$, $(d-1)^2$, $d(d-2)$ or $d(d-1)$. Secondly, consider a curve $\delta\in M_g^{Pl}$ with $g=(d-1)(d-2)/2$ such that $Aut(\delta)$ has an element of "very large" order, in the sense that this element is of order $d^2-3d+3$, $(d-1)^2$, $d(d-2)$ or $d(d-1)$. Then we investigate the groups $G$ for which $\delta\in\widetilde{M_g^{Pl}(G)}$ and also we determine the locus $\widetilde{M_g^{Pl}(G)}$ in these situations. Moreover, we work with the same question when $Aut(\delta)$ has an element of "large" order $\ell d$, $\ell (d-1)$ or $\ell(d-2)$ with $\ell\geq 2$ an integer.