New Ideas for Resolution of Singularities in Arbitrary Characteristic

Tohsuke Urabe

Published 2000-06-09, updated 2010-06-18Version 2

This is the manuscript for Proceedings of International Conference and Workshop on Valuation Theory held at University of Saskachewan, Canada in 1999. I have succeeded in showing that any two-dimensional hypersurface singularities of germs of varieties in any characteristic can be resolved by iterated monoidal transformations with centers in smooth subvarieties. The new proof for the two-dimensional case depends on new ideas. Ideas are essentially different from Abhyankar's one in 1956 and Lipman's one in 1978. It seems to be possible to generalize the new proof into higher dimensional cases, if we add several ideas further. In this article I try to explain my new ideas rather than partial result I explained at the conference.