P-Rank and semi-stable reduction of curves

Abstract

Let R be a discrete complete valuation ring, with field of fractions K, and with algebraically closed residue field k of characteristic p > 0. Let X be a germ of an R-curve at an ordinary double point. Consider a finite Galois covering f: Y → X, whose Galois group G is a p-group, such that Y is normal, and which is étale above X k≔ x × rk . Asume that Y has a semi-stable model :→ Y over R, and let y be a closed point of Y. If the inertia subgroup I(y) at y is cyclic of order p n , we compute the p-rank of tf −1 (y) by using a result of Raynaud. In particular, we prove that this p-rank is bounded by p n −1 .