© 2016, Springer International Publishing. Let Mg be the moduli space of smooth, genus g curves over an algebraically closed field K of zero characteristic. Denote by Mg(G) the subset of Mg of curves δ such that G (as a finite non-trivial group) is isomorphic to a subgroup of Aut(δ) , the full automorphism group of δ, and let Mg(G) ~ be the subset of curves δ such that G≅ Aut (δ). Now, for an integer d≥ 4 , let MgPl be the subset of Mg 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 MgPl(G):=MgPl∩Mg(G) and MgPl(G)~:=Mg(G)~∩MgPl. In this paper, we study some aspects of the irreducibility of MgPl(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 MgPl(G)~. In particular, we introduce the concept of being equation strongly irreducible (ES-Irreducible) for which the locus MgPl(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 M3Pl(G)~, whenever non-empty, is ES-Irreducible. In this article, we prove that this phenomenon does not occur for any odd d≥ 5. More precisely, let Z/ mZ be the cyclic group of order m, we show that MgPl(Z/(d-1)Z)~ is not ES-Irreducible for any odd integer d≥ 5 , and the number of its irreducible components is at least two. Furthermore, we conclude the previous result when d = 6 for the locus M10Pl(Z/3Z)~. 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.
- Plane non-singular curves
- automorphism groups