# 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