Reduction in the Case of Imperfect Residue Fields

Let K be a complete field with a discrete valuation v, ring of integers OK , and maximal ideal (πK). Let k := OK/(πK) be the residue field, assumed to be separably closed of characteristic p ≥ 0. LetX/K be a smooth proper geometrically irreducible curve of genus g ≥ 1. Let X /OK denote a regular model of X/K. Let Xk = ∑v i=1 riCi be its special fiber, where Ci/k is an irreducible component of Xk of multiplicity ri. Let e(Ci) denote the geometric multiplicity of Ci (see [BLR], 9.1/3). In particular, e(Ci) = 1 if and only if Ci/k is geometrically reduced. Any reduced curve C/k is geometrically reduced when k is perfect. Associate to X /OK the field extension kX/k generated by the following three types of subfields: by the fields H(C,OC), where C is any irreducible component of Xk; by the fields of rationality of all points P such that P is the intersection point of two components of X red k ; and by the fields of rationality of all points Q that belong to geometrically reduced components and such that Q is not smooth.