Abstract

New progress in the local uniformization problem in positive characteristic and the valuative interpretation and applications of the problems in local algebraic geometry and complex analysis has increased the interest to study the old Riemann-Zariski manifold. In these talks, we introduce this space and survey several old and new results on this topic.

1. Introducing the Riemann-Zariski manifold via the resolution problem. The classification of valuations in dimension one; Riemann surface attached to a curve (according to Dedekind). The classification problem in dimension two (Several approaches to classify: sequences of point blow-ups, the dual graph of resolution, Newton-Puiseaux representation of valuations and its relation to the plane curve parametrization problem, MacLane key-polynomials).
2. The continuation of the dimension two case. Some generalizations of these classifications in higher dimensions. The semigroup of a valuation centered on a ring and its relation to the resolution problem. The problem of classifying the structure of value semigroups. The problem of constructing a valuation with a prescribed value semigroup.
3. Some applications: Analysis on the Riemann-Zariski manifold in dimension two (parameterizations, measures, operators), the Zariski theory of complete ideals, multiplier ideals, Lelong numbers, the Izumi theorem.




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