Number theory/Algebraic geometry
On the ordinariness of coverings of stable curves
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:YX 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:YηXη 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:YsXs 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:YX 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:YηXη 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:YsXs 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
Yang, Yu 1

1 Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
@article{CRMATH_2018__356_1_17_0,
     author = {Yang, Yu},
     title = {On the ordinariness of coverings of stable curves},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {17--26},
     publisher = {Elsevier},
     volume = {356},
     number = {1},
     year = {2018},
     doi = {10.1016/j.crma.2017.11.013},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2017.11.013/}
}
TY  - JOUR
AU  - Yang, Yu
TI  - On the ordinariness of coverings of stable curves
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 17
EP  - 26
VL  - 356
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2017.11.013/
DO  - 10.1016/j.crma.2017.11.013
LA  - en
ID  - CRMATH_2018__356_1_17_0
ER  - 
%0 Journal Article
%A Yang, Yu
%T On the ordinariness of coverings of stable curves
%J Comptes Rendus. Mathématique
%D 2018
%P 17-26
%V 356
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2017.11.013/
%R 10.1016/j.crma.2017.11.013
%G en
%F CRMATH_2018__356_1_17_0
Yang, Yu. 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. http://www.numdam.org/articles/10.1016/j.crma.2017.11.013/

[1] Bouw, I. The p-rank of curves and covers of curves, Luminy, 1998 (Progress in Mathematics), Volume vol. 187, Birkhäuser, Basel, Switzerland (2000), pp. 267-277

[2] Crew, R. Étale p-covers in characteristic p, Compos. Math., Volume 52 (1984), pp. 31-45

[3] Y. Hoshi, On the pro-p absolute anabelian geometry of proper hyperbolic curves, RIMS Preprint, 1860.

[4] Liu, Q.; Lorenzini, D. Models of curves and finite covers, Compos. Math., Volume 118 (1999), pp. 61-102

[5] Mochizuki, S. The geometry of the compactification of the Hurwitz scheme, Publ. Res. Inst. Math. Sci., Volume 31 (1995), pp. 355-441

[6] Mochizuki, S. The profinite Grothendieck conjecture for closed hyperbolic curves over number fields, J. Math. Sci. Univ. Tokyo, Volume 3 (1996), pp. 571-627

[7] Raynaud, M. p-Groupes et réduction semi-stable des courbes, The Grothendieck Festschrift, Vol. III, Progress in Mathematics, vol. 88, Birkhäuser Boston, Boston, MA, USA, 1990, pp. 179-197

[8] Raynaud, M. Mauvaise réduction des courbes et p-rang, C. R. Acad. Sci. Paris, Ser. I, Volume 319 (1994), pp. 1279-1282

[9] Raynaud, M. Revêtements de la droite affine en caractéristique p>0 et conjecture d'Abhyankar, Invent. Math., Volume 116 (1994), pp. 425-462

[10] Saïdi, M. p-Rank and semi-stable reduction of curves, C. R. Acad. Sci. Paris Ser. I, Volume 326 (1998), pp. 63-68

[11] Saïdi, M. p-Rank and semi-stable reduction of curves II, Math. Ann., Volume 312 (1998), pp. 625-639

[12] Tamagawa, A. Resolution of nonsingularities of families of curves, Publ. Res. Inst. Math. Sci., Volume 40 (2004), pp. 1291-1336

[13] Yang, Y. On the existence of non-finite coverings of stable curves over complete discrete valuation rings, Math. J. Okayama Univ. (2018) http://www.kurims.kyoto-u.ac.jp/~yuyang/papersandpreprints/WRNFA.pdf (in press, see also)

[14] Yang, Y. Formulas for local and global p-ranks of coverings of curves, 1863 http://www.kurims.kyoto-u.ac.jp/~yuyang/papersandpreprints/FLG.pdf (RIMS Preprint 1863, see also)

[15] Yang, Y. On the boundedness and graph-theoreticity of p-ranks of coverings of curves, 1864 http://www.kurims.kyoto-u.ac.jp/~yuyang/papersandpreprints/BG.pdf (RIMS Preprint 1864, see also)

Cited by Sources: