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

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 $\ell \ge 2$ an integer.

This paper is a recent version of chapter 2 of arXiv:1503.01149

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