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 a covering Y → X that is generically étale (i.e., étale 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 affrmative (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.
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Akio Tamagawa, Resolution of Nonsingularities of Families of Curves. Publ. Res. Inst. Math. Sci. 40 (2004), no. 4, pp. 1291–1336DOI 10.2977/PRIMS/1145475448