Relative inequality between genera. Appendix by Michel Matignon and Taoufik Youssefi: Good reduction and surjective morphisms
Let \((K,v_ 0)\) be an algebraically closed valued field and \(M/L\) be an extension of function fields of one variable over \(K\). Let \(\{v_ i\}\) be distinct valuations on \(L\) which extend \(v_ 0\) such that the residual extensions \(L_{v_ i}/K_{v_ 0}\) are also function fields of one variable. Let \(\{w_{i,j}\}\) be some continuations of the \(\{v_ i\}\) to \(M\). The main result of this paper is the following inequality between the genera of the functions fields: \[ g(M/K) - \sum_{i,j} g(M_{w_{ij}}/K_{v_ 0}) \geq g (L/K) - \sum_ i g(L_{v_ i}/K_{v_ 0}) \geq 0. \] The proof uses an induction on the rank of \(v_ 0\). The case of rank one is proved as a consequence of an inequality between Euler-Poincaré characteristic of singular algebraic curves in rigid geometry. An application of the theorem is that if \(M/K\) has good reduction, then \(L/K\) also has good reduction.
scientific article; zbMATH DE number 492767
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