Wild models of curves

Abstract

Let K be a complete discrete valuation field with ring of integers OK and algebraically closed residue field k of characteristic p > 0. Let X/K be a smooth proper geometrically connected curve of genus g > 0, withX(K) 6= ∅ if g = 1. Assume thatX/K does not have good reduction, and that it obtains good reduction over a Galois extension L/K of degree p. Let Y/OL be the smooth model of XL/L. Let H := Gal(L/K). In this article, we provide information on the regular model of X/K obtained by desingularizing the wild quotient singularities of the quotient Y/H . The most precise information on the resolution of these quotient singularities is obtained when the special fiber Yk/k is ordinary. As a corollary, we are able to produce for each odd prime p an infinite class of wild quotient singularities having pairwise distinct resolution graphs. The information on the regular model of X/K also allows us to gather insight into the p-part of the component group of the Neron model of the Jacobian of X .