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Oberseminar WiSe 2010/11 Zusammenfassung des Vortrags von Kai Arzdorf

An Algorithmic Approach to Semistable Reduction

When considering curves defined over local fields, one is often interested in having a semistable model; that is, a flat model with reduced geometric fibers that have only ordinary double points as singularities. The well-known theorem of Deligne and Mumford ensures that for smooth projective and absolutely irreducible curves such a model can always be found, provided one allows a finite separable extension of the base field.

Several different proofs of this theorem exist; some of them make use of rigid analytic geometry. We will talk about a new proof of the theorem, which also exploits rigid analytic methods. But in contrast to most of the proofs published so far, we rely solely on local arguments, which do not involve the global geometry of the curve. Our hope is that the new proof described here can serve as a basis for an efficient and practical algorithm to compute semistable models.