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Zariski–Riemann spaceIn algebraic geometry, a Zariski–Riemann space or Zariski space of a subring k of a field K is a locally ringed space whose points are valuation rings containing k and contained in K. They generalize the Riemann surface of a complex curve. Zariski–Riemann spaces were introduced by Zariski (1940, 1944) who (rather confusingly) called them Riemann manifolds or Riemann surfaces. They were named Zariski–Riemann spaces after Oscar Zariski and Bernhard Riemann by Nagata (1962) who used them to show that algebraic varieties can be embedded in complete ones. Local uniformization (proved in characteristic 0 by Zariski) can be interpreted as saying that the Zariski–Riemann space of a variety is nonsingular in some sense, so is a sort of rather weak resolution of singularities. This does not solve the problem of resolution of singularities because in dimensions greater than 1 the Zariski–Riemann space is not locally affine and in particular is not a scheme. DefinitionThe Zariski–Riemann space of a field K over a base field k is a locally ringed space whose points are the valuation rings containing k and contained in K. Sometimes the valuation ring K itself is excluded, and sometimes the points are restricted to the zero-dimensional valuation rings (those whose residue field has transcendence degree zero over k). If S is the Zariski–Riemann space of a subring k of a field K, it has a topology defined by taking a basis of open sets to be the valuation rings containing a given finite subset of K. The space S is quasi-compact. It is made into a locally ringed space by assigning to any open subset the intersection of the valuation rings of the points of the subset. The local ring at any point is the corresponding valuation ring. The Zariski–Riemann space of a function field can also be constructed as the inverse limit of all complete (or projective) models of the function field. ExamplesThe Riemann–Zariski space of a curveThe Riemann–Zariski space of a curve over an algebraically closed field k with function field K is the same as the nonsingular projective model of it. It has one generic non-closed point corresponding to the trivial valuation with valuation ring K, and its other points are the rank 1 valuation rings in K containing k. Unlike the higher-dimensional cases, the Zariski–Riemann space of a curve is a scheme. The Riemann–Zariski space of a surfaceThe valuation rings of a surface S over k with function field K can be classified by the dimension (the transcendence degree of the residue field) and the rank (the number of nonzero convex subgroups of the valuation group). Zariski (1939) gave the following classification:
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