Valuations, Deformations, and Toric Geometry

By B. Teissier. 2003

107 citations recorded by the source corpus.

A study of the relation between a noetherian local domain with a given valuation and its associated graded ring with respect to the valuation, which in some cases is an esentially toric variety, possibly of infinite embedding dimension, but of finite Krull dimension. After extension of the valuation to a suitable completion (whose existence is not established in the paper) the relation becomes much more precise, and suggests a possible way to establish local uniformization for an excellent equicharacteristic local domain with an algebraically closed residue field, by deforming a partial toric embedded resolution of the essentially toric spectrum of the associated graded ring.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/aa752a3024b562c243ef4c301ba3d0200f37a059

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In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

RESOLUTION OF SINGULARITIES OF THREEFOLDS IN POSITIVE CHARACTERISTIC II (2008, unknown): “Abhyankar’s techniques have been greatly simplified in [17] under the same assumption on k. Although several written programs have appeared in the recent years to prove resolution of singularities [30] [28] or the weaker local uniformization theorem [ 43 ] over a perfect ground field, none has been ”
Resolution of singularities of threefolds in positive characteristic. I.: Reduction to local uniformization on Artin–Schreier and purely inseparable coverings (2008, unknown): “Although several written programs have appeared in the recent years to prove resolution of singularities [30] [28] or the weaker local uniformization theorem [43] over a perfect ground ﬁeld, none has been completed to this date.”
The Valuative Tree (2002, background): “ of the valuation associated to a ﬁnite or inﬁnite STKP. That is, we describe the structure of the graded ring grνC(x)[y], and compute the three invariants rk(ν), rat.rk(ν) and tr.deg(ν). We refer to [T2] for general results in higher dimensions. Given a ﬁnite or inﬁnite STKP we denote by ν:= val[(U”
Monomial Ideals, Binomial Ideals, Polynomial Ideals (2005, unknown): “Now the faithful ﬂatness of the family of Proposition 7.2 implies that it can be deﬁned in A 1 ( k ) × A g +1 ( k ) by equations which are deformations of the equations of the monomial curve (see [T2], § 5, proof of 5.49).”
Toric geometry and the Semple–Nash modification (2009, background): “3 of [34], the rank of the image in kr×L of the matrix 〈m − n〉 is equal to c.”