The structure of spaces of valuations and the local uniformization problem
The problem of resolution of singularities is a major problem in algebraic geometry. Local uniformization can be seen as its local version. For varieties over fields of characteristic zero, local uniformization was proved by Zariski in 1940 and resolution of singularities was proved by Hironaka in 1964. For algebraic varieties over fields of positive characteristic both problems are open in dimension greater than 3. Zariski's idea to solve the resolution of singularities problem for an algebraic variety was to prove local uniformization for all valuations of the associated function field and use the compactness of the Zariski space of valuations to glue the solutions together and construct a global resolution. Hence, his approach deals with two aspects: proving local uniformization and using structural properties of spaces of valuations to glue the local solutions. In this thesis, we present our contribution to both aspects. In most of the successful cases, local uniformization was first proved for rank one valuations and then extended to the general case. Local uniformization can be stated as a property of a valuation centered at a local ring R. One of our contributions to the local uniformization problem (which is joint work with Spivakovsky) is that in order to prove local uniformization for valuations centered at local rings in a category M which is closed under taking homomorphic images, finitely generated birational extensions and localizations, it is enough to prove that rank one valuations centered at members of M admit local uniformization. We also obtain this reduction for different versions of local uniformization, for instance, for embedded local uniformization and inseparable local uniformization. Our proofs are particularly important because they do not depend on the nature of the category M. We also work with henselian elements. Henselian elements are roots of polynomials appearing in Hensel's Lemma. We summarize unpublished results of Kuhlmann, van den Dries and Roquette to obtain that for a finite field extension (F|L,v), if F is contained in the absolute inertia field of L, then the valuation ring OF of (F,v) is generated as an OL-algebra by henselian elements. Moreover, we obtain a list of equivalent conditions under which OF is generated over OL by finitely many henselian elements. We prove that if the chain of prime ideals of OL is well-ordered by inclusion, then these conditions are satisfied. We give an example of a finite inertial extension (F|L,v) for which OF is not a finitely generated OL-algebra. We also present a theorem with a simple proof that relates the problem of local uniformization with the theory of henselian elements. This theorem shows, in particular, that if we obtain elimination of ramification for a function field for a good transcendence basis, then the valuation admits local uniformization. In our studies of spaces of valuations we define new topologies on spaces of valuations which extend naturally known topologies. We compare these topologies and show that in general they are not equal. We also obtain criteria under which the space of valuations taking values in a fixed ordered abelian group G is a closed subset in the space of all functions taking values in G. We study the works of Favre and Jonsson and Granja on the valuative tree. Favre and Jonsson prove that the set of all valuations centered at C[[x,y]] admits a tree structure, which they call the valuative tree. Granja extends this result to any two-dimensional regular local ring. In both works, the definition of non-metric rooted tree is not satisfactory. This is because the definition does not guarantee the existence of an infimum for any non-empty set of valuations. This infimum is necessary in order to define and study many concepts related to such trees. We give a more general definition of a rooted non-metric tree and prove that the set of all valuations satisfies this more general definition, namely, we prove that every non-empty set of valuations centered at a two-dimensional regular domain admits an infimum. We also generalize some topological results related to a non-metric tree, for instance that the weak tree topology is always coarser than the metric topology given by any parametrization.
Valuations, Local uniformization, Valuative tree, Spaces of valuations.
Doctor of Philosophy (Ph.D.)
Mathematics and Statistics