Resolution of Nonsingularities of Families of Curves

Abstract

In the present paper, we consider the following problem: For a given closed point x of a special fiber of a generically smooth family X → S of stable curves (with dim(S )=1 ), is there ac overingY → X that is genericallyetale (i.e., ´ etale over the generic fiber(s) of X → S, not only over the generic point(s) of X), where Y is also a family of stable curves, such that the image in X of the non-smooth locus of Y contains x? Among other things, we prove that this is affirmative (possibly after replacing S by a finite extension) in the case where S is the spectrum of a discrete valuation ring of mixed characteristic whose residue field is algebraic over Fp.