On the locus of smooth plane curves with a fixed automorphism group

TL;DR

This paper investigates the irreducibility of the moduli space of smooth plane curves with fixed automorphism groups, showing that for certain groups and degrees, the space is not represented by a single normal form, indicating multiple components.

Contribution

It proves that for odd degrees greater than 4, the locus of plane curves with cyclic automorphism groups is not ES-Irreducible, revealing multiple irreducible components, extending previous results to new cases.

Findings

The locus is not ES-Irreducible for odd degrees ≥5.

Number of irreducible components is at least two.

Results hold over algebraically closed fields of characteristic p > (d-1)(d-2)+1.

Abstract

In this paper, we study some aspects of the irreducibility of $\widetilde{M_g^{Pl}(G)}$ and its interrelation with the existence of "normal forms", i.e. non-singular plane equations (depending on a set of parameters) such that a specialization of the parameters gives a certain non-singular plane model associated to the elements of $\widetilde{M_g^{Pl}(G)}$. In particular, we introduce the concept of being equation strongly irreducible (ES-Irreducible) for which the locus $\widetilde{M_g^{Pl}(G)}$ is represented by a single "normal form". Henn, and Komiya-Kuribayashi, observed that $\widetilde{M_3^{Pl}(G)}$ is ES-Irreducible. In this paper we prove that this phenomena does not occur for any odd $d>4$. More precisely, let $\mathbb{Z}/m\mathbb{Z}$ be the cyclic group of order $m$, we prove that $\widetilde{M_g^{Pl}(\mathbb{Z}/(d-1)\mathbb{Z})}$ is not ES-Irreducible for any odd integer $d\geq5$, and the number of its irreducible components is at least two. Furthermore, we conclude the previous result when $d=6$ for the locus $\widetilde{M_{10}^{Pl}(\mathbb{Z}/3\mathbb{Z})}$. Lastly, we prove the analogy of these statements when $K$ is any algebraically closed field of positive characteristic $p$ such that $p>(d-1)(d-2)+1$.