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Given an algebraic variety X defined over an algebraically closed field, we study the space RZ(X,x) consisting of all the valuations of the function field of X which are centered in a closed point x of X. We concentrate on its homeomorphism type. We prove that, when x is a regular point, this homeomorphism type only depends on the dimension of X. If x is a singular point of a normal surface, we show that it only depends on the dual graph of a good resolution of (X,x) up to some precise equivalence. This is done by studying the relation between RZ(X,x) and the normalized non-Archimedean link of x in X coming from the point of view of Berkovich geometry. We prove that their behavior is the same., 31 pages

In the 1920s, B. N. Delaunay proved that the dual graph of the Voronoi diagram of a discrete set of points in a Euclidean space gives rise to a collection of simplices, whose circumspheres contain no points from this set in their interior. Such Delaunay simplices tessellate the convex hull of these points. An equivalent formulation of this property is that the characteristic functions of the Delaunay simplices form a partition of unity. In the paper this result is generalized to the so-called Delaunay configurations. These are defined by considering all simplices for which the interiors of their circumspheres contain a fixed number of points from the given set, in contrast to the Delaunay simplices, whose circumspheres are empty. It is proved that every family of Delaunay configurations generates a partition of unity, formed by the so-called simplex splines. These are compactly supported piecewise polynomial functions which are multivariate analogs of the well-known univariate B-splines. It is also shown that the linear span of the simplex splines contains all algebraic polynomials of degree not exceeding the degree of the splines.

Let p be normal singularity of the 2-dimensional Stein space V. Let π: M → V be a minimal good resolution of V, such that the irreducible components A i of A = π ―1 (p) are nonsingular and have only normal crossings. Associated to A is weighted dual graph Γ which, along with the genera of the A i , fully describes the topology and differentiable structure of A and the topological and differentiable nature of the embedding of A in M. In this paper we give the complete classification of weighted dual graphs which have only complete intersection singularities but no hypersurface singularities associated to them. We also give the complete classification of weighted dual graphs which have only complete intersection singularities associated with them.

the circuits of any finite graph G define a matroid. We call this the circuitmatroid and its dual the bond-matroid of G. In the present paper we determine a necessary and sufficient condition, in terms of matroid structure, for a given matroid M to be graphic (cographic), that is the bond-matroid (circuitmatroid) of some finite graph. The condition is that M shall be regular and shall not contain, in a sense to be explained, the circuit-matroid (bondmatroid) of a Kuratowski graph, that is a graph with one of the structures shown in Figure I. Some of the intermediate results seem to be of interest in themselves. These include the theory of dual matroids in ?2, Theorem (7.3) on regular matroids, and Theorem (8.4) on the bond-matroids of graphs. Our main theorem is evidently closely related to the theorem of Kuratowski on planar graphs [1]. This states that a graph is planar (i.e., can be imbedded in the plane) if and only if it contains no graph with the point set structure of a Kuratowski graph. Indeed it is not difficult to prove Kuratowski's Theorem from ours, using the principle that a graph is planar if and only if it has a dual graph, that is if and only if its circuit-matroid is graphic. However this paper is long enough already and we refrain from adding matter not essential to the proof and understanding of the main theorem. 2. Dual matroids. We define a matroid M on a set M, its flats and their dimensions as in HI, ?1. We call the dimension of the largest flat (M) also the dimension dM of the matroid. (See HI, ?2, for the notation (S)). We write ae(S) for the number of elements of any finite set S. We proceed to give a definition of the dual of M analogous to the definition of a dual vector space in terms of orthogonality. First, in analogy with A, ?4, we define a dendroid of M as a minimal sub