Skip to main content
SpringerLink
Log in

Loci of curves with subcanonical points in low genus

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Inside the moduli space of curves of genus three with one marked point, we consider the locus of hyperelliptic curves with a marked Weierstrass point, and the locus of non-hyperelliptic curves with a marked hyperflex. These loci have codimension two. We compute the classes of their closures in the moduli space of stable curves of genus three with one marked point. Similarly, we compute the class of the closure of the locus of curves of genus four with an even theta characteristic vanishing with order three at a certain point. These loci naturally arise in the study of minimal dimensional strata of Abelian differentials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves. Vol. I, volume 267 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, New York (1985)

  2. Bergström, J.: Cohomology of moduli spaces of curves of genus three via point counts. J. Reine Angew. Math. 622, 155–187 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Bainbridge, M., Möller, M.: The Deligne–Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3. Acta Math. 208(1), 1–92 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, D., Coskun, I.: Extremal higher codimension cycles on moduli spaces of curves. Proc. Lond. Math. Soc. 111(1), 181–204 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, D.: Square-tiled surfaces and rigid curves on moduli spaces. Adv. Math. 228(2), 1135–1162 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, D.: Strata of abelian differentials and the Teichmüller dynamics. J. Mod. Dyn. 7(1), 135–152 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, D., Möller, M.: Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus. Geom. Topol. 16(4), 2427–2479 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cornalba, M.: Moduli of curves and theta-characteristics. In: Lectures on Riemann Surfaces (Trieste, 1987), pp. 560–589. World Sci. Publ., Teaneck (1989)

  9. Caporaso, L., Sernesi, E.: Recovering plane curves from their bitangents. J. Algebra Geom. 12(2), 225–244 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cukierman, F.: Families of Weierstrass points. Duke Math. J. 58(2), 317–346 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. Diaz, S.: Exceptional Weierstrass points and the divisor on moduli space that they define. Mem. Amer. Math. Soc 56(327), iv+69 (1985)

    MathSciNet  MATH  Google Scholar 

  12. Dolgachev, I., Kanev, V.: Polar covariants of plane cubics and quartics. Adv. Math. 98(2), 216–301 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Eisenbud, D., Harris, J.: Irreducibility of some families of linear series with Brill–Noether number \(-1\). Ann. Sci. École Norm. Sup. (4) 22(1), 33–53 (1989)

    MathSciNet  MATH  Google Scholar 

  14. Faber, C.: Chow rings of moduli spaces of curves. I. The Chow ring of \(\overline{\cal M}_3\). Ann. Math. (2) 132(2), 331–419 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  15. Faber, C.: Chow rings of moduli spaces of curves. II. Some results on the Chow ring of \(\overline{\cal M}_4\). Ann. Math. (2) 132(3), 421–449 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. Farkas, G.: The birational type of the moduli space of even spin curves. Adv. Math. 223(2), 433–443 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Farkas, G.: Brill–Noether geometry on moduli spaces of spin curves. In: Classification of Algebraic Varieties, EMS Ser. Congr. Rep., pp. 259–276. Eur. Math. Soc., Zürich (2011)

  18. Faber, C., Pandharipande, R.: Relative maps and tautological classes. J. Eur. Math. Soc. (JEMS) 7(1), 13–49 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Faber, C., Pagani, N.: The class of the bielliptic locus in genus 3. Preprint, arXiv:1206.4301. To appear in Int. Math. Res. Not. (2015)

  20. Farkas, G., Verra, A.: The geometry of the moduli space of odd spin curves. Ann. Math. (2) 180(3), 927–970 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Getzler, E.: Topological recursion relations in genus 2. In: Integrable Systems and Algebraic Geometry (Kobe/Kyoto, 1997), pp. 73–106. World Sci. Publ., River Edge (1998)

  22. Getzler, E., Looijenga, E.: The Hodge polynomial of \(\overline{\cal M}_{3,1}\). Preprint, arXiv:math/9910174 (1999)

  23. Hain, R.: Normal functions and the geometry of moduli spaces of curves. In: Farkas, G., Morrison, I. (eds.) Handbook of Moduli, vol. I of Advanced Lectures in Mathematics. Somerville: International Press (2013)

  24. Harris, J., Mumford, D.: On the Kodaira dimension of the moduli space of curves. Invent. Math. 67(1), 23–88 (1982). With an appendix by William Fulton

    Article  MathSciNet  MATH  Google Scholar 

  25. Jensen, D.: Birational contractions of \(\overline{M}_{3,1}\) and \(\overline{M}_{4,1}\). Trans. Amer. Math. Soc. 365(6), 2863–2879 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kontsevich, M., Zorich, A.: Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Invent. Math. 153(3), 631–678 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Logan, A.: The Kodaira dimension of moduli spaces of curves with marked points. Amer. J. Math. 125(1), 105–138 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Scorza, G.: Sopra le curve canoniche di uno spazio lineare qualunque e sopra certi loro covarianti quartici. Atti R. Accad. Sci. Torino 35, 765–773 (1900)

    Google Scholar 

  29. Tarasca, N.: Brill–Noether loci in codimension two. Compos. Math. 149(9), 1535–1568 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tarasca, N.: Double total ramifications for curves of genus 2. Int. Math. Res. Not. 19, 9569–9593 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Montserrat, TiB: The divisor of curves with a vanishing theta-null. Compos. Math. 66(1), 15–22 (1988)

    MathSciNet  MATH  Google Scholar 

  32. Yang, S.: Calculating intersection numbers on moduli spaces of curves. Preprint, arXiv:0808.1974v2 (2010)

  33. Zorich, A.: Flat surfaces. In: Frontiers in Number Theory, Physics, and Geometry. I, pp. 437–583. Springer, Berlin (2006)

Download references

Acknowledgments

We would like to thank Izzet Coskun, Joe Harris, Scott Mullane, and Anand Patel for helpful conversations on related topics. We are grateful to the referee for many valuable suggestions for improving the exposition of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Boston College, Chestnut Hill, MA, 02467, USA

    Dawei Chen

  2. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA

    Nicola Tarasca

Authors
  1. Dawei Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicola Tarasca
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicola Tarasca.

Additional information

During the preparation of this article the first author was partially supported by NSF Grant DMS-1200329 and NSF CAREER Grant DMS-1350396.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, D., Tarasca, N. Loci of curves with subcanonical points in low genus. Math. Z. 284, 683–714 (2016). https://doi.org/10.1007/s00209-016-1670-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-016-1670-5

Keywords

Mathematics Subject Classification

Search

Navigation