Sobre el problema de resolución de singularidades en característica arbitraria.

Resumen:

The subject of this talk is the problem of resolution of singularities in algebraic geometry, but it is intended for a general mathematical audience. The problem of resolution of singularities asks whether, given an algebraic variety X over a field k, there exists a non-singular algebraic variety X0 and a proper map X0 → X which is one-to-one over the non-singular locus of X. If we cover X0 by affine charts, the problem becomes one of parametrizing pieces of X by
small pieces of the Euclidean space k^n. If char k = 0, resolution of singularities was constructed by H. Hironaka in 1964. This local version of the problem, called Local Uniformization, is usually stated in terms of valuations and can be interpreted as follows. Let (R, M, k) be a local k-algebra essentially of finite type without zero divisors) and let Rν be a valuation ring containing R and having the
same field of fractions as R. Find a smooth finite type R-algebra R0 such that R0 ⊂ Rν. The Local Uniformization Theorem asserts the existence of such an R0 ; it was proved by O. Zariski in 1940 in the case when char k=0 and is one of the central open problems in the field when char k=p > 0.
We will explain in detail the notion of the Zariski–Riemann space, formed by valuations centered at points of a given algebraic variety and Zariski’s program for using it in the proof of resolution of singularities. Time permitting, we will briefly discuss the difficulties that arise in trying to generalize Hironaka’s result to fields of positive characteristic.

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