Seminar
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| Location: | SLMath: Eisenbud Auditorium, Online/Virtual |
Why is resolution of singularities so hard in characteristic p?
Local uniformization is the process of resolving the singularities of a variety locally along a fixed valuation. The essential step is to reduce multiplicity along the valuation. This method was introduced by Zariski in the 1930s. He proved local uniformization in characteristic zero and all dimensions and deduced resolution of singularities of varieties in characteristic zero and dimension less than or equal to three. Zariski finds a “hypersurface of maximal contact” to prove reduction of multiplicity, leading to local uniformization. Hironka proved resolution of singularities in characteristic zero in the 1960s. The critical result on which the massive edifice of his proof is built is the simple fact that hypersurfaces of maximal contact always exist in characteristic zero. The established proofs of resolution of singularities in positive characteristic are in dimension less than or equal to three, by Abhyankar and most generally by Cossart and Piltant. These proofs first prove local uniformization. The essential difference in positive characteristic is that hypersurfaces of maximal contact may not exist.
We discuss Zariski’s proof of reduction of multiplicity in characteristic zero, and isolate where the proof fails in characteristic p. Zariski’s proof and most proofs of resolution of singularities proceed by realizing the local ring of a singularity as a quasi-finite extension of a regular local ring. In local uniformization, invariants of the extension of the valuation are used to help achieve a reduction of multiplicity. In characteristic zero, these invariants are very simple and control the extension, but in characteristic p, defect can occur which leads to chaos in the extension. The behavior of the extension of valuations in the extension of rings is controlled by the associated graded ring along the valuation. We discuss the structure of this extension of associated graded rings and the effect of defect on it. The structure is very simple when there is no defect, but is extremely complicated if there is defect. We use this analysis to show that Zariski’s algorithm succeeds in proving reduction of multiplicity if there is no defect in the extension.
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