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Stable reduction of finite covers of curves

Published online by Cambridge University Press:  13 January 2006

Qing Liu
Qing Liu
Affiliation:
CNRS, Laboratoire A2X, Institut de Mathématiques de Bordeaux, Université de Bordeaux I, 351, cours de la Libération, 33405 Talence, FranceQing.Liu@math.u-bordeaux1.fr
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Abstract

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Let K be the function field of a connected regular scheme S of dimension 1, and let $f : X\to Y$ be a finite cover of projective smooth and geometrically connected curves over K with $g(X)\ge 2$. Suppose that f can be extended to a finite cover ${\mathcal X} \to {\mathcal Y}$ of semi-stable models over S (it is known that this is always possible up to finite separable extension of K). Then there exists a unique minimal such cover. This gives a canonical way to extend $X\to Y$ to a finite cover of semi-stable models over S.

Keywords

curvesfinite coversstable reduction14G2014H3014D1011G20
Type
Research Article
Information
Compositio Mathematica , Volume 142 , Issue 1 , January 2006 , pp. 101 - 118
Copyright
Foundation Compositio Mathematica 2006