Comptes Rendus
no. 1
Number theory/Algebraic geometry
On the ordinariness of coverings of stable curves

Presented by: Claire Voisin

Yu Yang1
1 Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 17-26.

In the present paper, we study the ordinariness of coverings of stable curves. Let f:Y→X be a morphism of stable curves over a discrete valuation ring R with algebraically closed residue field of characteristic p>0. Write S for Spec R and η (resp. s) for the generic point (resp. closed point) of S. Suppose that the generic fiber Xη of X is smooth over η, that the morphism fη:Yη→Xη over η on the generic fiber induced by f is a Galois étale covering (hence Yη is smooth over η too) whose Galois group is a solvable group G, that the genus of the normalization of each irreducible component of the special fiber Xs is ≥2, and that Ys is ordinary. Then we have that the morphism fs:Ys→Xs over s induced by f is an admissible covering. This result extends a result of M. Raynaud concerning the ordinariness of coverings to the case where Xs is a stable curve. If, moreover, we suppose that G is a p-group, and that the p-rank of the normalization of each irreducible component of Xs is ≥2, we can give a numerical criterion for the admissibility of fs.

Dans la présente Note, nous étudions l'ordinarité des revêtements de courbes stables. Soit f:Y→X un morphisme de courbes stables sur un anneau de valuation discrète R, dont le corps résiduel est algébriquement clos, de caractéristique p>0. Notons S pour Spec(R) et η (resp. s) le point générique (resp. le point fermé) de S. Supposons que la fibre générique Xη de X est lisse au-dessus de η, que le morphisme fη:Yη→Xη des fibres génériques induit par f au-dessus de η soit un revêtement étale galoisien (et donc Yη est aussi lisse au-dessus de η), dont le groupe de Galois G est résoluble, que le genre des normalisations des composantes irréductibles de la fibre spéciale Xs soit au moins 2 et que Ys soit ordinaire. Alors, le morphisme fs:Ys→Xs induit par f au-dessus de s est un revêtement admissible. Ce résultat étend un énoncé de M. Raynaud sur l'ordinarité des revêtements lorsque Xs est une courbe stable. Si, de plus, on suppose que G est un p-groupe et que le p-rang de la normalisation de chaque composante irréductible de Xs est au moins 2, nous pouvons donner un critère numérique pour l'admissibilité de fs.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2017.11.013

Yu Yang 1

1 Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan 
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Yu Yang. On the ordinariness of coverings of stable curves. Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 17-26. doi : 10.1016/j.crma.2017.11.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.11.013/

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