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manuscripta mathematica

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

Local invariants attached to Weierstrass points

  • Robin de JongEmail author
Open Access
Article

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.

Mathematics Subject Classification (2000)

14H55 14H10 14G40 

Notes

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.

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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of LeidenLeidenThe Netherlands

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