Uniform approximation of Abhyankar valuation ideals in smooth function fields

Ein, Lawrence; Lazarsfeld, Robert; Smith, Karen E.

doi:10.1353/ajm.2003.0010

Cited by 109 publications

(144 citation statements)

References 23 publications

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“…Let X be a normal projective klt variety and L an ample line bundle on X. Then there exists a constant C = C(X, L) such that Their proofs, which appear at the end of this section, use multiplier ideals and take inspiration from [DEL00] and [ELS03].…”

Section: Uniform Fujita Approximationmentioning

confidence: 99%

Thresholds, valuations, and K-stability

Blum

Jönsson

2020

Advances in Mathematics

117

181

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Let X be a normal complex projective variety with at worst klt singularities, and L a big line bundle on X. We use valuations to study the log canonical threshold of L, as well as another invariant, the stability threshold. The latter generalizes a notion by Fujita and Odaka, and can be used to characterize when a Q-Fano variety is K-semistable or uniformly K-stable. It can also be used to generalize volume bounds due to Fujita and Liu. The two thresholds can be written as infima of certain functionals on the space of valuations on X. When L is ample, we prove that these infima are attained. In the toric case, toric valuations acheive these infima, and we obtain simple expressions for the two thresholds in terms of the moment polytope of L.arXiv:1706.04548v1 [math.AG] 14 Jun 2017for all positive integers p, q. This, together with the Noetherianity of X, implies that {J (X, (c/p) · a p )} p∈N has a unique maximal element that is called the c-th asymptotic multiplier ideal and denoted by J (X, c · a • ). Note that J (X, c · a • ) = J (X, (c/p) · a p ) for all p divisible enough.Asymptotic multiplier ideals also satisfy a subadditivity property. See [Laz04, Theorem 11.2.3] for the case when X is smooth.Corollary 5.6. Let a • be a graded sequence of ideals on X. If m ∈ N * and c ∈ Q * + , thenNext we give a containment relation for the multiplier ideal associated to the graded sequence of valuation ideals. The result appears in [ELS03] in the case when v is divisorial.Proof. It is an immediate consequence of the valuative criterion for membership in the multiplier ideal [BdFFU15, Theorem1.2] that J (X, c · a • (v)) ⊂ a cv(a•(v))−A(v) (v).Since v(a • (v)) = 1 (see [Blu16b, Lemma 3.5]), the proof is complete.

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Section: Uniform Fujita Approximationmentioning

confidence: 99%

Thresholds, valuations, and K-stability

Blum

Jönsson

2020

Advances in Mathematics

117

181

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“…real-valued valuations on X for which equality holds in the Abhyankar inequality (1.1), see [ELS,Proposition 4.8].…”

Section: Definition 13mentioning

confidence: 99%

Valuations and Plurisubharmonic Singularities

Boucksom

Favre²,

Jönsson³

2008

Publ. Res. Inst. Math. Sci.

129

179

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We extend to higher dimensions some of the valuative analysis of singularities of plurisubharmonic (psh) functions developed by the first two authors. Following Kontsevich and Soibelman we describe the geometry of the space V of all normalized valuations on C[x 1 , . . . , x n ] centered at the origin. It is a union of simplices naturally endowed with an affine structure. Using relative positivity properties of divisors living on modifications of C n above the origin, we define formal psh functions on V, designed to be analogues of the usual psh functions. For bounded formal psh functions on V, we define a mixed Monge-Ampère operator which reflects the intersection theory of divisors above the origin of C n . This operator associates to any (n − 1)-tuple of formal psh functions a positive measure of finite mass on V. Next, we show that the collection of Lelong numbers of a given germ u of a psh function at all infinitely near points induces a formal psh functionû on V. When ϕ is a psh Hölder weight in the sense of Demailly, the generalized Lelong number ν ϕ (u) equals the integral of u against the Monge-Ampère measure ofφ. In particular, any generalized Lelong number is an average of valuations. We also show how to compute the multiplier ideal of u and the relative type of u with respect to ϕ in the sense of Rashkovskii, in terms ofû andφ.

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“…For a valuation v with zero-dimensional center on an n-dimensional variety X, the volume was defined in [21] Boucksom-Küronya-MacLean-Szemberg [7] show that the limit…”

Section: Valuation Ideals and Volumementioning

confidence: 99%

“…21 [10]. The cluster K of centers of v ξ,t can be easily described from the continued fraction expansion t = n 1 consists of s = n i centers; if t = n 1 then they all lie on the proper transform of the germ Γ : y = ξ(x) ,…”

mentioning

confidence: 99%

Nagata type statements

Roé

Supino

2018

Banach Center Publ.

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Nagata solved Hilbert's 14-th problem in 1958 in the negative. The solution naturally lead him to a tantalizing conjecture that remains widely open after more than half a century of intense efforts. Using Nagata's theorem as starting point, and the conjecture, with its multiple variations, as motivation, we explore the important questions of finite generation for invariant rings, for support semigroups of multigraded algebras, and for Mori cones of divisors on blown up surfaces, and the rationality of Waldschimdt constants. Finally we suggest a connection between the Mori cone of the Zariski-Riemann space and the continuity of the Waldschmidt constant as a function on the space of valuations.These notes correspond to the course of the same title given by the first author in the workshop "Asymptotic invariants attached to linear series" held in the Pedagogical University of Cracow from May 16 to 20, 2016.

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Uniform approximation of Abhyankar valuation ideals in smooth function fields

Cited by 109 publications

References 23 publications

Thresholds, valuations, and K-stability

Thresholds, valuations, and K-stability

Valuations and Plurisubharmonic Singularities

Nagata type statements

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