On the ordinariness of coverings of stable curves

Abstract

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 SpecR 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.