Stable modification of relative curves
By M. Temkin. 2007
62 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/9edeee874c632ae1650202e7716d8f042655f9b6
How researchers see this paper
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In Applied Mathematics
- Sam Payne
- Matthew Baker
- Joseph Rabinoff
- Laura Demarco
- Mattias Jonsson
- Robert Rumely
- Ruochuan Liu
- Kiran S. Kedlaya
In Geometry and Topology
- Matthew Baker
- Sam Payne
- Joseph Rabinoff
- David Jensen
- M. Temkin
- Omid Amini
- Dhruv Ranganathan
- Lucia Caporaso
Citations in the corpus
- On the structure of non-archimedean analytic curves (2013, background): “(Conversely, one can study the local structure of an analytic curve directly and derive the semistable reduction theorem: see [25].”
- Relative Riemann-Zariski spaces (2008, methodology): “Our improvement to the stable modification theorem [ Tem2 , 1.1] is in the control on the base change one has to perform in order to construct a stable modification of a relative curve C → S. Namely, we prove that in order to find a stable modification of a relative curve with semi-stable U-fibers i”