Construction of coverings of pointed curves. (Construction de revêtements de courbes pointées.) (French) Zbl 1047.14013
From the introduction: On établit dans cet article le theoreme suivant:
Théorème. Soient \(k\) un corps fertile [“large field”, voir F. Pop, Ann. Math. (2) 144, No. 1, 1–34 (1996; Zbl 0862.12003)], \(\Gamma\) une courbe projective, lisse et connexe sur \(k\), \(\varepsilon\in\Gamma(k)\) un point rationnel, \(G\) un \(k\)-schéma en groupes fini étale, \(X\) un \(G\)-torseur sur \(k\). Alors il existe:
(i) une \(k\)-courbe projective, lisse et géométriquement connexe \(C\), et un \(k\)-morphisme \(\pi:C\to\Gamma\); (ii) une action fidèle de \(G\) sur \(C\), faisant de \(\pi\) un \(G\)-torseur au-dessus d’un ouvert de \(\Gamma\) contenant \(\varepsilon\);
(iii) un isomorphisme \(\pi^{-1}(\varepsilon)@>\sim>> X\) de \(G\)-torseurs.
Théorème. Soient \(k\) un corps fertile [“large field”, voir F. Pop, Ann. Math. (2) 144, No. 1, 1–34 (1996; Zbl 0862.12003)], \(\Gamma\) une courbe projective, lisse et connexe sur \(k\), \(\varepsilon\in\Gamma(k)\) un point rationnel, \(G\) un \(k\)-schéma en groupes fini étale, \(X\) un \(G\)-torseur sur \(k\). Alors il existe:
(i) une \(k\)-courbe projective, lisse et géométriquement connexe \(C\), et un \(k\)-morphisme \(\pi:C\to\Gamma\); (ii) une action fidèle de \(G\) sur \(C\), faisant de \(\pi\) un \(G\)-torseur au-dessus d’un ouvert de \(\Gamma\) contenant \(\varepsilon\);
(iii) un isomorphisme \(\pi^{-1}(\varepsilon)@>\sim>> X\) de \(G\)-torseurs.
MSC:
| 14H30 | Coverings of curves, fundamental group |
| 14M17 | Homogeneous spaces and generalizations |
| 14E20 | Coverings in algebraic geometry |
| 14H20 | Singularities of curves, local rings |
Citations:
Zbl 0862.12003References:
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