manuscripta mathematica

, Volume 78, Issue 1, pp 111–128 | Cite as

Inegalite relative des genres

  • Taoufik Youssefi
  • Michel Matignon
Article
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Abstract

Let (K, v0) be an algebrically closed valued field. Let M/L be an extension of function fields of one variable over K and {vi}1≤i≤s be distinct valuations on L which prolong v0 and have transcendental residue extensions (Lvi/Kv0). If {wj}1≤j≤t are prolongations of the {vi}1≤i≤s to M, we show the following inequality between the genera of the functions fields:
$$g(M/K) - \sum\limits_{1 \leqslant j \leqslant t} {g(Mw_j /Kv_0 ) \geqslant g(L/K)} - \sum\limits_{1 \leqslant i \leqslant s} {g(Lv_i /Kv_0 ) \geqslant 0.} $$
. As an application we show that if M/K has good reduction, L/K also has good reduction. This result generalizes a result of H. Lange [L]. In the appendix we give other “known” results related to Lange's theorem.
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Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Taoufik Youssefi
    • 1
  • Michel Matignon
    • 2
  1. 1.Faculté des Sciences Département de MathématiquesUniversité Ibn TofailKénitraMaroc
  2. 2.U.R.A. 226, Université Bordeaux I CEREMABTalence, CédexFrance

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