Resolution of nonsingularities of families of curves
Let \(S\) be a 1-dimensional, noetherian, normal, integral, separated scheme and denote by \(K\) the function field of \(S\). A pair \((S', K')\) is a finite extension of \((S,K)\) if \(K'\) is a finite extension of \(K\) and \(S'\) is the integral closure of \(S\) in \(K\) (hence \(K'\) coincides with the function field of \(S'\)). In the present paper the author studies the following problem: for a given closed point \(x\) of a special fiber of a generically smooth family \(X \rightarrow S\) of stable curves is there a covering \(Y \rightarrow X\) that is generically étale, 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 many results, the author proves the following. Assume that \(S\) is strictly of mixed characteristic, and that the residue field of each closed point of \(S\) is algebraic over \(\mathbb{F}_p\), for some \(p > 0\). Let \((X^*, D)\) be a generically smooth, stable marked curve over \(S\). Then for each closed point \(x \in X^*\) there exists a finite extension \((S', K')\) of \((S,K)\) and a generically tame stable covering \((Y^*_{S'}, E_{S'})\) of \((X^*_{S'}, D_{S'})\) over \(S'\) such that the image in \(X^*\) of the vertical components in the fibers of \(Y^*_{S'} \rightarrow X^*_{S'}\) contains \(x\) (in particular the image of the non-smooth locus of \(Y^*_{S'}\) in \(X^*\) contains \(x\)).
scientific article; zbMATH DE number 2185483