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Specializations of coverings in characteristic \(p>0\). (Spécialisation des revêtements en caractéristique \(p>0\).) (French) Zbl 0999.14004

Based on the abstract: Let \(G\) be a finite group with a \(p\)-Sylow subgroup \(Q\) of order \(p\), and let \(R\) be a complete discrete valuation ring with fraction field \(K\) of characteristic zero and algebraically closed residue field of characteristic \(p\), and let \(e\) be the ramification index of \(R\). \(X\) is the projective line over \(R\), fix the three sections \(0\), \(1\) and \(\infty\). Let \(Y_K \rightarrow X_K\) be a finite normal Galois cover with Galois group \(G\), which is geometrically connected and ramified exactly over the three sections \(0\), \(1\) and \(\infty\). It is shown that if \(e < \frac{p-1}{n}\), then \(Y_K\) has good reduction over \(R\). More precisely, the integral closure \(Y\) of \(X\) is smooth over \(R\) and tamely ramified on \(X\) above \(0\), \(1\) and \(\infty\).

MSC:

14E22 Ramification problems in algebraic geometry
14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
13F30 Valuation rings
14G20 Local ground fields in algebraic geometry
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References:

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