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 $X\to S$ of stable curves (with $\dim (S)=1$), is there a covering $Y\to X$ that is generically étale (i.e., étale over the generic fiber(s) of $X\to S$, 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 ${\mathbb{F}}_p$.