Stable modification of relative curves

Details Stable modification of relative curves Stable modification of relative curves

2007-07-26

arxiv.org/abs/0707.3953v3

Mathematics

Algebraic Geometry

60 pages, third version, the paper was revised due to referee's report, section 2 was divided into sections 2 and 6, to appear

Scientific paper

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.

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