Discrete valuations centered on local domains

Uniform bounds and symbolic powers on smooth varieties

“…Using the subadditivity theorem of [3], we prove In the case of symbolic powers it is elementary to check that J ( q (e) ) ⊆ q, so Theorem A follows from the "abstract" Theorem B. As another application, we establish a result (Theorem 2.5) rendering effective and extending in certain directions a theorem of Izumi [9], [7] dealing with ideals arising from a valuation.…”

“…Then we get a graded family of ideals o • = {o k } on X by putting o k = ν * O Y (−kD). Note that this includes the symbolic powers q (k) in (iii) as a special case, as well as the graded family of ideals associated to an m-valuation on X in the sense of [7].…”

Uniform approximation of Abhyankar valuation ideals in smooth function fields

“…More generally, Theorem A can be applied to a composite of valuations to yield the following: Thus even though the algebra ℓ∈N π * O X (−mℓD) (3) 1 We adopt the convention that a m−e = R when m < e. 2 Izumi's statement actually deals with less general valuations but more general rings. It is proved in [12], [22], and [11].…”

Generalization of a result of Hickel, Ito and Izumi about a Diophantine inequality

“…On the other hand, every m-valuation can be realized as a Rees valuation of an m-primary ideal. Therefore, if ν is an m-valuation we obtain, by using [16,17] or [2,9], the linear equivalence between the m-adic and the ν-adic topologies. Hence, when R is supposed analytically irreducible, there exist s > 0, t > 0 and c > 0 such that for all x ∈ R, …”