manuscripta mathematica

, Volume 129, Issue 3, pp 273–292 | Cite as

Local invariants attached to Weierstrass points

  • Robin de JongEmail author
Open Access


Let X/S be a hyperelliptic curve of genus g over the spectrum of a discrete valuation ring. Two fundamental numerical invariants are attached to X/S: the valuation d of the hyperelliptic discriminant of X/S, and the valuation δ of the Mumford discriminant of X/S (equivalently, the Artin conductor). For a residue field of characteristic 0 as well as for X/S semistable the invariants d and δ are known to satisfy certain inequalities. We prove an exact formula relating d and δ with intersection theoretic data determined by the distribution of Weierstrass points over the special fiber, in the semistable case. We also prove an exact formula for the stable Faltings height of an arbitrary curve over a number field, involving local contributions associated to its Weierstrass points.

Mathematics Subject Classification (2000)

14H55 14H10 14G40 


The author thanks the Mittag-Leffler Institute in Djursholm for its hospitality during a visit. He thanks Sylvain Maugeais for several discussions related to the theme of this paper, Lidia Stoppino for pointing out the reference [2], and the anonymous referee for a number of helpful remarks. The author is supported by VENI-grant 639.031.619 of the Netherlands Organisation for Scientific Research (NWO).

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Arakelov S.Y.: Families of algebraic curves with fixed degeneracies. Math. USSR Izvestija 5, 1277–1302 (1971)CrossRefGoogle Scholar
  2. 2.
    Bost J.-B.: Semi-stability and heights of cycles. Invent. Math. 118, 223–253 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bost, J.-B., Mestre, J.-F., Moret-Bailly, L.: Sur le calcul explicite des “classes de Chern” des surfaces arithmétiques de genre 2. In: Séminaire sur les pinceaux de courbes elliptiques, vol. 183, pp. 69–105. Astérisque (1990)Google Scholar
  4. 4.
    Burnol J.-F.: Weierstrass points on arithmetic surfaces. Invent. Math. 107, 421–432 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cornalba M., Harris J.: Divisor classes associated to families of stable varieties, with applications to the moduli space of curves. Ann. Sci. Ec. Num. Sup. 21, 455–475 (1988)zbMATHMathSciNetGoogle Scholar
  6. 6.
    de Jong R.: Arakelov invariants of Riemann surfaces. Doc. Math. 10, 311–329 (2005)zbMATHMathSciNetGoogle Scholar
  7. 7.
    de Jong R.: Explicit Mumford isomorphism for hyperelliptic curves. J. Pure Appl. Algebra 208, 1–14 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Deligne, P.: Le déterminant de la cohomologie. In: Contemporary Mathematics, vol. 67, pp. 93–177. American Mathematical Society (1987)Google Scholar
  9. 9.
    Deligne P., Mumford D.: The irreducibility of the space of curves of given genus. Publ. Math. de l’I.H.E.S. 36, 75–110 (1969)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Faltings G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73, 349–366 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Faltings G.: Calculus on arithmetic surfaces. Ann. Math. 119, 387–424 (1984)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Guàrdia J.: Analytic invariants in Arakelov theory for curves. C.R. Acad. Sci. Paris Ser. I 329, 41–46 (1999)zbMATHGoogle Scholar
  13. 13.
    Hain R., Reed D.: On the Arakelov geometry of moduli spaces of curves. J. Differ. Geom. 67, 195–228 (2004)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Kausz I.: A discriminant and an upper bound for ω 2 for hyperelliptic arithmetic surfaces. Compos. Math. 115, 37–69 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Laudal, O.A., Lønsted, K.: Deformations of curves I. Moduli for hyperelliptic curves. Lect. Notes in Math., vol. 687, pp. 150–167. Springer (1978)Google Scholar
  16. 16.
    Lockhart P.: On the discriminant of a hyperelliptic curve. Trans. Am. Math. Soc. 342, 729–752 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Lønsted K., Kleiman S.L.: Basics on families of hyperelliptic curves. Compos. Math. 38, 83–111 (1979)zbMATHGoogle Scholar
  18. 18.
    Liu Q.: Conducteur et discriminant minimal de courbes de genre 2. Compos. Math. 94, 51–79 (1994)zbMATHGoogle Scholar
  19. 19.
    Liu, Q.: Algebraic Geometry and Arithmetic Curves. Oxford Graduate Texts in Mathematics, vol. 6. Oxford Science Publications, Oxford (2002)Google Scholar
  20. 20.
    Matsusaka S.: Some numerical invariants of hyperelliptic fibrations. J. Math. Kyoto Univ. 30, 33–57 (1990)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Maugeais S.: Relèvement des revêtements p-cycliques des courbes rationnelles semi-stables. Math. Ann. 327, 365–393 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Moret-Bailly, L.: Métriques permises. In: Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, vol. 127, pp. 29–87. Astérisque (1985)Google Scholar
  23. 23.
    Moret-Bailly L.: La formule de Noether pour les surfaces arithmétiques. Inv. Math. 98, 491–498 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Moret-Bailly, L.: Hauteurs et classes de Chern sur les surfaces arithmétiques. In: Séminaire sur les pinceaux de courbes elliptiques, vol. 183, pp. 37–58. Astérisque (1990)Google Scholar
  25. 25.
    Mumford D.: Stability of projective varieties. l’Ens. Math. 23, 33–100 (1977)MathSciNetGoogle Scholar
  26. 26.
    Namikawa Y., Ueno K.: The complete classification of fibres in pencils of curves of genus two. Manuscripta math. 9, 143–186 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Ogg A.P.: On pencils of curves of genus two. Topology 5, 353–367 (1966)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Ogg A.P.: Elliptic curves and wild ramification. Am. J. Math. 89, 1–21 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Saito T.: Conductor, discriminant, and the Noether formula of arithmetic surfaces. Duke Math. J. 57, 151–173 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Saito T.: The discriminants of curves of genus 2. Compos. Math. 69, 229–240 (1989)zbMATHGoogle Scholar
  31. 31.
    Szpiro, L.: Un peu d’effectivité. In: Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, vol. 127, pp. 275–287. Astérisque (1985)Google Scholar
  32. 32.
    Szpiro, L.: Sur les propriétés numériques du dualisant relatif d’une surface arithmétique. In: The Grothendieck Festschrift, Volume III. Progress in Mathematics, vol. 88. Birkhäuser Verlag, Basel (1990)Google Scholar
  33. 33.
    Ueno, K.: Discriminants of curves of genus two and arithmetic surfaces. Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata, Kinokuniya, Tokyo, pp. 749–770 (1987)Google Scholar
  34. 34.
    Viehweg E.: Canonical divisors and the additivity of the Kodaira dimension for morphisms of relative dimension one. Compos. Math. 35, 197–223 (1977)zbMATHMathSciNetGoogle Scholar
  35. 35.
    Yamaki K.: Cornalba-Harris equality for semistable hyperelliptic curves in positive characteristic. Asian J. Math. 8, 409–426 (2004)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Zhang, S.-W.: Gross-Schoen cycles, dualising sheaves, and triple product L-series. PreprintGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of LeidenLeidenThe Netherlands

Personalised recommendations