I‑AIMS Institute for Artificial Intelligence
and Mathematical Sciences
/

On the generic curve of genus 3

Tanush Shaska, Jennifer L. Thompson

I-AIMS paper 2003-1 · Published in Contemp. Math. 369 (2005) · 2005

We study genus g coverings of full moduli dimension of degree d = [(g+3)/2]. There is a homomorphism between the corresponding Hurwitz space ℋ of such covers to the moduli space ℳ_g of genus g curves. In the case g = 3, using the signature of such covering we provide an equation for the generic ternary quartic. Further, we discuss the degenerate subloci of the corresponding Hurwitz space of such covers from the computational group theory viewpoint. In the last section, we show that one of these degenerate loci corresponds to the locus of curves with automorphism group C₃. We give necessary conditions in terms of covariants of ternary quartics for a genus 3 curve to belong to this locus.

MSC classes: Primary 14H10, 14H45; Secondary 14Q05

PDF

References

  1. [Be] W. Baily, On the automorphism group of a generic curve of genus > 2, J. Math. Kyoto Univ. 1 (1961/1962), 101–108; correction, 325.
  2. [Co2] M. Coppens The Weierstrass gap sequence of the ordinary ramification points of trigonal coverings of P ; existence of a kind of Weierstrass gap sequence. J. Pure Appl. Algebra 43 (1986), no. 1, 11–25.
  3. [Co3] M. Coppens The Weierstrass gap sequences of the total ramification points of trigonal coverings of P . Nederl. Akad. Wetensch. Indag. Math. 47 (1985), no. 3, 245–276.
  4. [Di] J. Dixmier, On the projective invariants of quartic plane curves. Adv. in Math. 64 (1987), no. 3, 279–304.
  5. [MSV] K. Magaard, S. Shpectorov, H. Voelklein, A GAP package for braid orbit computation, and applications. Experimental Math. (to appear).
  6. [MV] K. Magaard and H. Völklein, The general curve covers P with monodromy group An. (preprint)
  7. [MS] K. Magaard, T. Shaska, S. Shpectorov, and H. Volklein, The locus of curves with prescribed automorphism group. Communications in arithmetic fundamental groups (Kyoto, 1999/2001). S¯urikaisekikenky¯usho K¯oky¯uroku No. 1267 (2002), 112–141.
  8. [Mu] D. Mumford, The Red Book on Varieties and Schemes, Lect. Notes in Math. 1358, Springer, 1999.
  9. [FL] C. Faber and E. Looijenga, Moduli of curves and abelian varieties. The Dutch Intercity Seminar on Moduli. Aspects of Mathematics, E33. Friedr. Vieweg & Sohn, Braunschweig, 1999.
  10. [GS] J. Gutierrez and T. Shaska, Hyperelliptic curves with extra involutions, Int. Math. Res. Notices, 2004 (submitted).
  11. [KK] A. Kuribayashi, K. Komiya, On the structure of the automorphism group of a compact Riemann surface of genus 3. Bull. Fac. Sci. Engrg. Chuo Univ. 23 (1980), 1–34.
  12. [KSV] V. Krishnamorthy, T. Shaska, H. Völklein, Invariants of binary forms, Developments in Mathematics, Kluwer Academic Publishers, Boston, MA, 2004, (in press).
  13. [Sh] T. Shioda, On the graded ring of invariants of binary octavics. Amer. J. Math. 89 1967 1022–1046.
  14. [Oh] T. Ohno, Application of representation theory of SL(3) to invariant elements of ternary quartics. Bull. Fac. Sci. Eng. Chuo Univ. 43 (2000), 7–16 (2001).
  15. [Ma] T. Maeda, On the invariant field of binary octavics. Hiroshima Math. J. 20 (1990), no. 3, 619–632.
  16. [Ts] Sh. Tsuyumine, On Siegel modular forms of degree three. Amer. J. Math. 108 (1986), no. 4, 755–862.
  17. [Sh1] T. Shaska, Curves of genus 2 with (n, n)-decomposable Jacobians, J. Symbolic Comput. 31 (2001), no. 5, 603–617.
  18. [Sh2] T. Shaska, Genus 2 curves with (3,3)-split Jacobian and large automorphism group, Algorithmic Number Theory (Sydney, 2002), 6, 205-218, Lect. Not. in Comp. Sci., 2369, Springer, Berlin, 2002.
  19. [Sh3] T. Shaska, Computational aspects of hyperelliptic curves, Computer mathematics. Proceedings of the sixth Asian symposium (ASCM 2003), Beijing, China, April 17-19, 2003. River Edge, NJ: World Scientific. Lect. Notes Ser. Comput. 10, 248-257 (2003).
  20. [Sh4] 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.
  21. [Sh5] T. Shaska, Genus 2 curves with degree 3 elliptic subcovers, Forum. Math., vol. 16, 2, pg. 263-280, 2004.
  22. [Sh6] T. Shaska, Some special families of hyperelliptic curves, J. Algebra Appl., vol 3, No. 1 (2004), 75-89.
  23. [Sh8] T. Shaska, Field of moduli of algebraic curves, (work in progress).
  24. [Sh9] T. Shaska, Hyperelliptic curves of genus 3 with split Jacobian, (preprint)
  25. [SV] T. Shaska and H. Völklein, Elliptic subfields and automorphisms of genus two fields, Algebra, Arithmetic and Geometry with Applications. Papers from Shreeram S. Abhyankar’s 70th Birthday Conference (West Lafayette, 2000), pg. 687 - 707, Springer (2004).
  26. [SV2] T. Shaska and H. Völklein, Genus 2 curves with degree 5 elliptic subcovers, (submitted).
  27. [V] H. Völklein, Groups as Galois groups. An introduction. Cambridge Studies in Advanced Mathematics, 53. Cambridge University Press, Cambridge, 1996. 300 Brink Hall, Department of Mathematics, University of Idaho, Moscow, ID, 83844.

All preprints