Overweight deformations of affine toric varieties and local uniformization

Abstract

Given an equicharacteristic complete noetherian local domain R with algebraically closed residue field k, we first present a combinatorial proof of embedded local uniformization for zero-dimensional valuations of R whose associated graded ring gr ν R with respect to the filtration defined by the valuation is a finitely generated k-algebra. The main idea here is that some of the birational toric maps which provide embedded pseudo-resolutions for the affine toric variety corresponding to gr ν R also provide local uniformizations for ν on R. These valuations are necessarily Abhyankar (for zero-dimensional valuations this means that the value group is Z r with r = dimR). In a second part we show that conversely, given an excellent noetherian equicharacteristic local domain R with algebraically closed residue field, if the zero-dimensional valuation ν of R is Abhyankar, there are local domains R ′ which are essentially of finite type over R and dominated by the valuation ring Rν (ν-modifications of R) such that the semigroup of values of ν on R ′ is finitely generated, and therefore so is the k-algebra gr ν R ′ . Combining the two results and using the fact that Abhyankar valuations behave well under completion gives a proof of local uniformization for rational Abhyankar valuations and, by a specialization argument, for all Abhyankar valuations.As a by-product we obtain a description of the valuation ring of a rational Abhyankar valuation as an inductive limit indexed by N of birational toric maps of regular local rings. One of our main tools, the valuative Cohen theorem, is then used to study the extensions of rational monomial Abhyankar valuations of the ring k[[x 1 , . . . , xr]] to monogenous integral extensions and the nature of their key polynomials. In the conclusion we place the results in the perspective of local embedded resolution of singularities by a single toric modification after an appropriate re-embedding. 1

The analogue of Izumi's Theorem for Abhyankar valuations

“…More precisely, he proved in this case that an Abhyankar valuation is quasi-monomial after a sequence of blow-ups. Teissier proved a similar statement for a complete equicharacteristic local domain R when the residue field of the Abhyankar valuation is equal to the residue field of R and is algebraically closed (see [17,Proposition 6.24]). Thus in these cases, by Theorem 1.3, the ν-adic and the m-adic topologies are linearly equivalent.…”

Ramification of local rings along valuations

“…(3), strong monomialization holds for Abhyankar valuations in positive characteristic, as follows from [9], (a strong form of local uniformization is proven for Abhyankar valuations by Knaf and Kuhlmann), and thus Proposition 1.3 holds in positive characteristic. A description of gr ν (R) for ν an Abhyankar valuation dominating a (singular) local ring R, over an algebraically closed field of arbitrary characteristic, and a proof of local uniformization for Abhyankar valuations derived from this construction, has been recently given by Teissier in [11].…”

On the homeomorphism type of some spaces of valuations

“…This was a key point in his program for resolution. It turns out to be also a key result in some recent attempts to solve this problem in positive characteristic following new strategies also using local uniformization (see [2,12]). …”