Stable modification of relative curves
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.
Cite
@article{arxiv.0707.3953,
title = {Stable modification of relative curves},
author = {Michael Temkin},
journal= {arXiv preprint arXiv:0707.3953},
year = {2010}
}
Comments
60 pages, third version, the paper was revised due to referee's report, section 2 was divided into sections 2 and 6, to appear in JAG