Limits of equisymmetric $1$-complex dimensional families of Riemann surfaces
DOI:
https://doi.org/10.7146/math.scand.a-26306Abstract
We describe the limit surfaces of some equisymmetric $1$-complex dimensional families of Riemann surfaces in the boundary of the Deligne-Mumford compactification of moduli space. We provide a description of such nodal Riemann surfaces in terms of the group of automorphisms defining the family. We apply our method to some known examples.
References
Abikoff, W., Degenerating families of Riemann surfaces, Ann. of Math. (2) 105 (1977), no. 1, 29–44. https://doi.org/10.2307/1971024
Bers, L., On spaces of Riemann surfaces with nodes, Bull. Amer. Math. Soc. 80 (1974), 1219–1222. https://doi.org/10.1090/S0002-9904-1974-13686-4
Bers, L., Deformations and moduli of Riemann surfaces with nodes and signatures, Math. Scand. 36 (1975), 12–16. https://doi.org/10.7146/math.scand.a-11557
Birkenhake, C. and Lange, H., A family of abelian surfaces and curves of genus four, Manuscripta Math. 85 (1994), no. 3-4, 393–407. https://doi.org/10.1007/BF02568206
Buser, P., Geometry and spectra of compact Riemann surfaces, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2010, reprint of the 1992 edition. https://doi.org/10.1007/978-0-8176-4992-0
Costa, A. F., Izquierdo, M., and Parlier, H., Connecting $p$-gonal loci in the compactification of moduli space, Rev. Mat. Complut. 28 (2015), no. 2, 469–486. https://doi.org/10.1007/s13163-014-0161-7
Costa, A. F., Izquierdo, M., and Ying, D., On Riemann surfaces with non-unique cyclic trigonal morphism, Manuscripta Math. 118 (2005), no. 4, 443–453. https://doi.org/10.1007/s00229-005-0593-y
Costa, A. F., Izquierdo, M., and Ying, D., On the family of cyclic trigonal Riemann surfaces of genus 4 with several trigonal morphisms, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 101 (2007), no. 1, 81–86.
Costa, A. F., Izquierdo, M., and Ying, D., On cyclic $p$-gonal Riemann surfaces with several $p$-gonal morphisms, Geom. Dedicata 147 (2010), 139–147. https://doi.org/10.1007/s10711-009-9444-4
Deligne, P. and Mumford, D., The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969), no. 36, 75–109.
Dryden, E. B. and Parlier, H., Collars and partitions of hyperbolic cone-surfaces, Geom. Dedicata 127 (2007), 139–149. https://doi.org/10.1007/s10711-007-9172-6
Edge, W. L., A pencil of specialized canonical curves, Math. Proc. Cambridge Philos. Soc. 90 (1981), no. 2, 239–249. https://doi.org/10.1017/S0305004100058692
Hubbard, J. H. and Koch, S., An analytic construction of the Deligne-Mumford compactification of the moduli space of curves, J. Differential Geom. 98 (2014), no. 2, 261–313.
Macbeath, A. M. and Singerman, D., Spaces of subgroups and Teichmüller space, Proc. London Math. Soc. (3) 31 (1975), no. 2, 211–256. https://doi.org/10.1112/plms/s3-31.2.211
Magaard, K., Shaska, T., Shpectorov, S., and Völklein, H., The locus of curves with prescribed automorphism group, Sūrikaisekikenkyūsho Kōkyūroku (2002), no. 1267, “Communications in arithmetic fundamental groups (Kyoto, 1999/2001)'', 112–141.
Matsumoto, Y. and Montesinos-Amilibia, J. M., Pseudo-periodic maps and degeneration of Riemann surfaces, Lecture Notes in Mathematics, vol. 2030, Springer, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-22534-5
Miranda, R., Graph curves and curves on $K3$ surfaces, in “Lectures on Riemann surfaces (Trieste, 1987)'', World Sci. Publ., Teaneck, NJ, 1989, pp. 119--176.
Rodríguez, R. E. and González-Aguilera, V., Fermat's quartic curve, Klein's curve and the tetrahedron, in “Extremal Riemann surfaces (San Francisco, CA, 1995)'', Contemp. Math., vol. 201, Amer. Math. Soc., Providence, RI, 1997, pp. 43--62. https://doi.org/10.1090/conm/201/02634
Wootton, A., The full automorphism group of a cyclic $p$-gonal surface, J. Algebra 312 (2007), no. 1, 377–396. https://doi.org/10.1016/j.jalgebra.2007.01.018
