On the Locus of Smooth Plane Curves with a Fixed Automorphism Group

Abstract

Let M _g be the moduli space of smooth, genus g curves over an algebraically closed field K of zero characteristic. Denote by M _g (G) the subset of M _g of curves ${\delta}$ such that G (as a finite non-trivial group) is isomorphic to a subgroup of Aut${(\delta)}$, the full automorphism group of ${\delta}$, and let ${\widetilde{M_g(G)}}$ be the subset of curves ${\delta}$ such that ${G\cong {\rm Aut}(\delta)}$. Now, for an integer ${d\geq 4}$, let ${M_g^{Pl}}$ be the subset of M _g representing smooth, genus g curves that admit a non-singular plane model of degree d (in such case, ${g=(d-1)(d-2)/2}$), and consider the sets ${M_g^{Pl}(G):=M_g^{Pl} \cap M_g(G)}$ and ${\widetilde{M_g^{Pl}(G)}:=\widetilde{M_g(G)} \cap M_g^{Pl}}$.

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 (Die Automorphismengruppen dar algebraischen Functionenkorper vom Geschlecht 3. Inagural-dissertation, Heidelberg, 1976), and Komiya-Kuribayashi (On Weierstrass points and automorphisms of curves of genus three. In: Algebraic geometry (Proc. Summer Meeting, Copenhagen 1978), LNM, vol 732. Springer, New York 1979), observed that ${\widetilde{M_3^{Pl}(G)}}$, whenever non-empty, is ES-Irreducible. In this article, we prove that this phenomenon does not occur for any odd ${d\geq5}$. More precisely, let ${\mathbb{Z}/m\mathbb{Z}}$ be the cyclic group of order m, we show 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 ${p > (d-1)(d-2)+1}$.