# Resolution of Nonsingularities of Families of Curves

### Akio Tamagawa

Kyoto University, Japan

## 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 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 **F***p*.

## Cite this article

Akio Tamagawa, Resolution of Nonsingularities of Families of Curves. Publ. Res. Inst. Math. Sci. 40 (2004), no. 4, pp. 1291–1336

DOI 10.2977/PRIMS/1145475448