algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, a Zariski–Riemann space or Zariski space of a

subring In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...

''k'' of a field ''K'' is a locally ringed space whose points are

valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every non-zero element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''. Given a field ''F'', if ''D'' is a subring of ' ...

s containing ''k'' and contained in ''K''. They generalize the Riemann surface of a complex curve. Zariski–Riemann spaces were introduced by who (rather confusingly) called them Riemann manifolds or Riemann surfaces. They were named Zariski–Riemann spaces after Oscar Zariski and

Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...

by 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 In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, which is a non-singular variety ''W'' with a Proper morphism, proper birational map ''W''→''V''. For varieties ov ...

. 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.

Definition

The Zariski–Riemann space of a field ''K'' over a base field ''k'' is a locally ringed space whose points are the

s 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 In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of ...

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.

Examples

The Riemann–Zariski space of a curve

The 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 surface

The 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). gave the following classification: *Dimension 2. The only possibility is the trivial valuation with rank 0, valuation group 0 and valuation ring ''K''. *Dimension 1, rank 1. These correspond to divisors on some blowup of ''S'', or in other words to divisors and infinitely near points of ''S''. They are all discrete. The center in ''S'' can be either a point or a curve. The valuation group is Z. *Dimension 0, rank 2. These correspond to germs of algebraic curves through a point on a normal model of ''S''. The valuation group is isomorphic to Z+Z with the

lexicographic order In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a ...

. *Dimension 0, rank 1, discrete. These correspond to germs of non-algebraic curves (given for example by ''y''= a non-algebraic formal

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...

in ''x'') through a point of a normal model. The valuation group is Z. *Dimension 0, rank 1, non-discrete, value group has incommensurable elements. These correspond to germs of transcendental curves such as ''y''=''x''^π through a point of a normal model. The value group is isomorphic to an ordered group generated by 2 incommensurable real numbers. *Dimension 0, rank 1, non-discrete, value group elements are commensurable. The value group can be isomorphic to any dense subgroup of the rational numbers. These correspond to germs of curves of the form ''y''=Σ''a''_''n''''x''^''b''_''n'' where the numbers ''b''_''n'' are rational with unbounded denominators.

References

* * * * * {{DEFAULTSORT:Zariski-Riemann space Algebraic geometry Bernhard Riemann