Universidad de Chile, Santiago de ChileSome results about the moduli space Mg and Ag
Slides
Moduli spaces of Riemann surfaces Mg and of abelian varieties Ag are central objects of research in Complex Geometry. Since they are related by the Torelli map, it is natural to search for results about Ag using what is known for Mg. This approach is particularly fruitful when considering group actions. There are famous bounds for several concepts regarding the geometry of group actions on compact Riemann surfaces of genus g ≥ 2: The well known Hurwitz bound 84(g − 1), which is attained for infinite genera, and also in infinitely many others not. Another remarkable bounds are Wiman’s bound 4g + 2, which is attained for every g [7], and the Accola-Maclachlan bound 8g + 8 [1, 4]. Following Accola-Maclachlan idea, other geometrically meaningful bounds are obtained; such as 4g + 1 [2] and 4g − 4 [5]. In this talk we will explain the question that motivates the search of these bounds and discuss extensions of these ideas.
Specifically, we consider a compact Riemann surface X of genus g ≥ 2 and T ∈ Aut(X) an automorphism of large primer order q > g. It is known that either q = 2g + 1, and this gives Lefschetz surfaces, or q = g + 1. We [6] are interested in the understanding of Riemann surfaces, and in their corresponding Jacobian varieties, in this second case. That is to provide a classification of those Riemann surfaces and some descriptions of the corresponding Jacobian varieties, in terms of decompositions and other properties. In particular, we compute the Riemann matrix of the Accola-Maclachlan curve [1, 4] of genus 4.
Part of this work is in collaboration with Sebastián Reyes-Carocca, from Universidad de Chile.
References
[1] R. Accola, On the number of automorphisms of a closed Riemann surface, Trans. Am. Math. Soc. 131 (1968), 398–408.
[2] A. F. Costa and M. Izquierdo, One-dimensional families of Riemann surfaces of genus g with 4g + 4 automorphisms, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 112 (2018), no. 3, 623–631.
[3] M. Izquierdo, M., S. Reyes-Carocca, A.M. Rojas, On families of Riemann surfaces with automorphisms, Journal of Pure and Applied Algebra 225, in press, pp.1-21. https://doi.org/10.1016/j.jpaa.2021.106704
[4] C. Maclachlan, A bound for the number of automorphisms of a compact Riemann surface, J. London Math. Soc. 44 (1969), 265–272.
[5] S. Reyes-Carocca, On compact Riemann surfaces of genus g with 4g − 4 automorphisms, Israel J. Math. 237 (2020), 415–436.
[6] S. Reyes-Carocca and A. M. Rojas., On large prime actions on Riemann surfaces, J. Group Theory 25 (2022), 887–940, DOI 10.1515/jgth-2020-0140.
[7] A. Wiman, Ueber die hyperelliptischen Curven und diejenigen von Geschlechte p = 3, welche eindeutige Transformationen in sich zulassen, Bihang till K. Svenska Vet.-Akad. Handlingar, Stockholm 21 (1895-6), 1–28.
