Stable modification of relative curves

TL;DR

This paper extends semi-stable modification theorems for families of curves, showing that after certain base alterations, any family admits a minimal semi-stable model characterized by specific fiber properties.

Contribution

It generalizes Deligne-Mumford and de Jong's theorems by proving the existence of minimal semi-stable modifications after base alterations for broader classes of curve families.

Findings

Existence of minimal semi-stable modifications after base alterations.

Characterization of these modifications by fiber properties.

Application of uniformization and Riemann-Zariski spaces to the proof.

Abstract

We generalize theorems of Deligne-Mumford and de Jong on semi-stable modifications of families of proper curves. The main result states that after a generically \'etale alteration of the base any (not necessarily proper) family of multipointed curves with semi-stable generic fiber admits a minimal semi-stable modification. The latter can also be characterized by the property that its geometric fibers have no certain exceptional components. The main step of our proof is uniformization of one-dimensional extensions of valued fields. Riemann-Zariski spaces are then used to obtain the result over any integral base.