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Stable reduction of curves and tame ramification

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Abstract

We study stable reduction of curves in the case where a tamely ramified base extension is sufficient. If X is a smooth curve defined over the fraction field of a strictly henselian discrete valuation ring, there is a criterion, due to Saito, that describes precisely, in terms of the geometry of the minimal model with strict normal crossings of X, when a tamely ramified extension suffices in order for X to obtain stable reduction. For such curves we construct an explicit extension that realizes the stable reduction, and we furthermore show that this extension is minimal. We also obtain a new proof of Saito’s criterion, avoiding the use of ℓ-adic cohomology and vanishing cycles.

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Authors and Affiliations

  1. Institut für Algebraische Geometrie, Gottfried Wilhelm Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    Lars Halvard Halle

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  1. Lars Halvard Halle
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Correspondence to Lars Halvard Halle.

Additional information

This work was partly done during the Moduli Spaces-year 2006/2007 at Institut Mittag-Leffler (Djursholm, Sweden).

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Halle, L.H. Stable reduction of curves and tame ramification. Math. Z. 265, 529–550 (2010). https://doi.org/10.1007/s00209-009-0528-5

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  • DOI: https://doi.org/10.1007/s00209-009-0528-5

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