arXiv:math/0205314v1 [math.AG] 30 May 2002 The locus of curves with prescribed automorphism group April 24, 2002 K. Magaard 1 , T.Shaska, S. Shpectorov 2 , and H. V¨olklein 3 Wayne State University, University of California at Irvine, Bowling Green State University and University of Florida Abstract: Let G be a finite group, and g ≥ 2. We study the locus of genus g curves that admit a G-action of given type, and inclusions between such loci. We use this to study the locus of genus g curves with prescribed automorphism group G. We completely classify these loci for g = 3 (including equations for the corresponding curves), and for g ≤ 10 we classify those loci corresponding to “large” G. 1 Introduction There is a vast literature on automorphism groups of compact Riemann surfaces, beginning in the 19th century with Schwartz, Klein, Hurwitz, Wiman and others. However, most of the literature is quite recent. In the first part of the paper, we survey the main results. By covering space theory, a finite group G acts (faithfully) on a genus g curve if and only if it has a genus g generating system (see section 5 below). Using this purely group-theoretic condition, Breuer [Br] classified all groups that act on a curve of genus ≤ 48. This was a major computational effort using the computer algebra system GAP [GAP]. It greatly improved on several papers dealing with small genus, by various authors (see the references in part I). Of course, for each group in Breuer’s list, all subgroups are also in the list. This raises the question how to pick out those groups that occur as the full automorphism group of a genus g curve. This question is answered in Part II of the paper. Let M g be the moduli space of genus g curves. We study the locus L in M g of curves admitting a G-action of given ramification type (resp., signature). All components of L have the same dimension which depends only on the signature of the G-action. Restricting the action to a subgroup H of G yields an inclusion of L into the corresponding locus L ′ for the action of H . If dim L < dim L ′ then for ”most” curves in L ′ , the H -action does not extend to a G-action of the type defining L. Thus one is led to classify the pairs (L, L ′ ) with dim L = dim L ′ . This is done in Lemma 4.1 below: It turns out that such pairs exist only in very restricted cases, in particular only if dim L≤ 3. From that we derive a necessary and sufficient condition for a group to occur as the full automorphism group of a genus g-curve (Theorem 5.3). After Part II was written, we found references to papers of Singerman [Si2] and Ries [Ri] which contain similar results. Their method is different, of analytic nature, using Teichm¨ uller theory and Fuchsian groups. Our approach is based on algebraic geometry, using the algebraic structure of Hurwitz spaces and the moduli spaces M g ; therefore, it can be used to obtain information on fields of definition. This aspect may be studied in later work. In Part III we apply the above criterion to the data compiled by Breuer (available from his website, see 1.5 below). Our first application is in the case of genus 3, which is already quite rich and shows the power of our group-theoretic method. We obtain the full list of automorphism groups in genus 3 plus equations for the corresponding curves. This result is scattered over several long papers by Kuribayashi and various co-authors [KuKo1], [KuKo2], [KK1], [KK2], some of which contain errors (see section 6 below) and some are not available in most libraries. None of them seems to give a complete account of this basic result. Here we show how to derive it quickly from the group-theoretic data. Further, we obtain the list of “large” groups Aut(X g ) (see 1.3 below for the definition) up to g = 10. Plus the dimension and number of components of the corresponding loci in M g , see Table 4. 1 Partially supported by NSA grant MDA904-01-1-0037 2 Partially supported by NSA grant MDA904-01-1-0023 3 Partially supported by NSF grant DMS-9970357 1