Equations for formal toric degenerations

Bernard H Teissier (1)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

Let $R$ be a complete equicharacteristic noetherian local domain and $\nu$ a valuation of its field of fractions whose valuation ring dominates $R$ with trivial residue field extension. The semigroup of values of $\nu$ on $R\setminus \{0\}$ is not finitely generated in general. We produce equations in an appropriate generalized power series ring for the algebra encoding the degeneration of $R$ to the toric graded algebra ${\rm gr}_\nu R$ associated to the filtration defined by $\nu$. We apply this to represent $\nu$ as the limit of a sequence of Abhyankar semivaluations (valuations on quotients) of $R$ with finitely generated semigroups.

Keywords: Toric degeneration; valuation theory; Local uniformization; 2000 Mathematics Subject Classification. 14M25 14E15 14B05 Toric geometry Valuations toric degeneration local uniformization; 2000 Mathematics Subject Classification. 14M25; 14E15; 14B05 Toric geometry; Valuations; toric degeneration; local uniformization

Domains: Mathematics [math]; Algebraic Geometry [math.AG]

Bernard H Teissier. Equations for formal toric degenerations. 2023. ⟨hal-03954995⟩