Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arakelov S.Y.: Families of algebraic curves with fixed degeneracies. Math. USSR Izvestija 5, 1277–1302 (1971)
Bost J.-B.: Semi-stability and heights of cycles. Invent. Math. 118, 223–253 (1994)
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)
Burnol J.-F.: Weierstrass points on arithmetic surfaces. Invent. Math. 107, 421–432 (1992)
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)
de Jong R.: Arakelov invariants of Riemann surfaces. Doc. Math. 10, 311–329 (2005)
de Jong R.: Explicit Mumford isomorphism for hyperelliptic curves. J. Pure Appl. Algebra 208, 1–14 (2007)
Deligne, P.: Le déterminant de la cohomologie. In: Contemporary Mathematics, vol. 67, pp. 93–177. American Mathematical Society (1987)
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)
Faltings G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73, 349–366 (1983)
Faltings G.: Calculus on arithmetic surfaces. Ann. Math. 119, 387–424 (1984)
Guàrdia J.: Analytic invariants in Arakelov theory for curves. C.R. Acad. Sci. Paris Ser. I 329, 41–46 (1999)
Hain R., Reed D.: On the Arakelov geometry of moduli spaces of curves. J. Differ. Geom. 67, 195–228 (2004)
Kausz I.: A discriminant and an upper bound for ω 2 for hyperelliptic arithmetic surfaces. Compos. Math. 115, 37–69 (1999)
Laudal, O.A., Lønsted, K.: Deformations of curves I. Moduli for hyperelliptic curves. Lect. Notes in Math., vol. 687, pp. 150–167. Springer (1978)
Lockhart P.: On the discriminant of a hyperelliptic curve. Trans. Am. Math. Soc. 342, 729–752 (1994)
Lønsted K., Kleiman S.L.: Basics on families of hyperelliptic curves. Compos. Math. 38, 83–111 (1979)
Liu Q.: Conducteur et discriminant minimal de courbes de genre 2. Compos. Math. 94, 51–79 (1994)
Liu, Q.: Algebraic Geometry and Arithmetic Curves. Oxford Graduate Texts in Mathematics, vol. 6. Oxford Science Publications, Oxford (2002)
Matsusaka S.: Some numerical invariants of hyperelliptic fibrations. J. Math. Kyoto Univ. 30, 33–57 (1990)
Maugeais S.: Relèvement des revêtements p-cycliques des courbes rationnelles semi-stables. Math. Ann. 327, 365–393 (2003)
Moret-Bailly, L.: Métriques permises. In: Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, vol. 127, pp. 29–87. Astérisque (1985)
Moret-Bailly L.: La formule de Noether pour les surfaces arithmétiques. Inv. Math. 98, 491–498 (1989)
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)
Mumford D.: Stability of projective varieties. l’Ens. Math. 23, 33–100 (1977)
Namikawa Y., Ueno K.: The complete classification of fibres in pencils of curves of genus two. Manuscripta math. 9, 143–186 (1973)
Ogg A.P.: On pencils of curves of genus two. Topology 5, 353–367 (1966)
Ogg A.P.: Elliptic curves and wild ramification. Am. J. Math. 89, 1–21 (1967)
Saito T.: Conductor, discriminant, and the Noether formula of arithmetic surfaces. Duke Math. J. 57, 151–173 (1988)
Saito T.: The discriminants of curves of genus 2. Compos. Math. 69, 229–240 (1989)
Szpiro, L.: Un peu d’effectivité. In: Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, vol. 127, pp. 275–287. Astérisque (1985)
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)
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)
Viehweg E.: Canonical divisors and the additivity of the Kodaira dimension for morphisms of relative dimension one. Compos. Math. 35, 197–223 (1977)
Yamaki K.: Cornalba-Harris equality for semistable hyperelliptic curves in positive characteristic. Asian J. Math. 8, 409–426 (2004)
Zhang, S.-W.: Gross-Schoen cycles, dualising sheaves, and triple product L-series. Preprint
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
de Jong, R. Local invariants attached to Weierstrass points. manuscripta math. 129, 273–292 (2009). https://doi.org/10.1007/s00229-009-0259-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-009-0259-2