I‑AIMS Institute for Artificial Intelligence
and Mathematical Sciences
/

Subvarieties of the hyperelliptic moduli determined by group actions

Tanush Shaska

I-AIMS paper 2006-4 · Published in Serdica Math. J. 32 (2006), 355–374 · 2006

Let ℋ_g be the moduli space of genus g hyperelliptic curves. In this note, we study the locus ℋ_g(G, σ) in ℋ_g of curves admitting a G-action of given ramification type σ and inclusions between such loci. For each genus we determine the list of all possible groups, the inclusions among the loci, and the corresponding equations of the generic curve in ℋ_g(G, σ). The proof of the results is based solely on representations of finite subgroups of PGL₂(ℂ) and the Riemann-Hurwitz formula.

MSC classes: 14Q05, 14Q15, 14R20, 14D22

PDF

References

  1. José Bertin, Compactification des schémas de Hurwitz, C.R. Acad. Sci. Paris I, vol. 322 (1996), 1063–1066.
  2. R. Brandt and H. Stichtenoch, Die Automorphismengrupenn Hyperelliptischer Kurven. Manuscripta Math 55 (1986), no. 1, 83–92.
  3. E. Bujalance, J. M. Gamboa, G. Gromadzki, The full automorphism groups of hyperelliptic Riemann surfaces, Manuscripta Math. 79 (1993), no. 3-4, 267–282.
  4. The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4; 2006, (http://www.gap-system.org).
  5. The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4; 2006, package SmallGroups, by E. A. O’Brien, B. Eick, and H. U. Besche (http://www-gap.dcs.st-and.ac.uk/ gap/Packages/sgl.html)
  6. J. Gutierrez, A polynomial decomposition algorithm over factorial domains, Comptes Rendues Mathematiques, de Ac. de Sciences, 13 (1991), 81-86.
  7. J. Gutierrez and T. Shaska, Hyperelliptic curves with extra involutions, LMS J. of Comput. Math., 8 (2005), 102-115.
  8. J. Gutierrez, D. Sevilla, and T. Shaska,, Hyperelliptic curves of genus 3 and their automorphisms, Lect. Notes Comp, vol 13.(2005),109–123.
  9. F. Klein, Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree. Dover Publications, Inc., New York, N. Y., 1956.
  10. E. Previato and T. Shaska, Vanishing theta nulls and algebraic curves with automorphisms, (work in progress).
  11. E. Previato and T. Shaska, Theta functions and automorphisms of curves, (work in progress).
  12. E. Previato, T. Shaska, and S. Wijesiri, Vanishing theta-nulls on genus two curves with split Jacobians, (work in progress).
  13. D. Sevilla and T. Shaska, Hyperelliptic curves with reduced automorphism group A5, Appl. Algebra Engrg. Comm. Comput., (to appear).
  14. T. Shaska, Some special families of hyperelliptic curves, J. Algebra Appl., vol 3, No. 1 (2004), 75-89.
  15. T. Shaska, Determining the automorphism group of hyperelliptic curves, Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, ACM Press, pg. 248 - 254, 2003.
  16. A. Weaver, Hyperelliptic surfaces and their moduli. Geom. Dedicata 103 (2004), 69–87.

All preprints