3-Fold Log Flips

Shokurov, V. V.

doi:10.1070/im1993v040n01abeh001862

Cited by 202 publications

(273 citation statements)

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“…Kawakita [16] used this ideal sheaf to prove inversion of adjunction on log canonicity and Shokurov [22] also used it implicitly.…”

Section: Introductionmentioning

confidence: 99%

A characteristic p analogue of plt singularities and adjoint ideals

Takagi

2007

Math. Z.

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Abstract. We introduce a new variant of tight closure and give an interpretation of adjoint ideals via this tight closure. As a corollary, we prove that a log pair (X, ∆) is plt if and only if the modulo p reduction of (X, ∆) is divisorially F-regular for all large p ≫ 0. Here, divisorially F-regular pairs are a class of singularities in positive characteristic introduced by Hara and Watanabe [10] in terms of Frobenius splitting. IntroductionThe multiplier ideal sheaf J (X, D) associated to a log pair (X, D) (i.e., X is a normal complex variety and D is an R-divisor on X) is defined in terms of resolution of singularities and discrepancy divisors, and one can view this ideal sheaf as measuring how singular the pair (X, D) is. However, when X is smooth and D is a (Cartier) integral divisor on X, the multiplier ideal sheaf J (X, D) is nothing but O X (−D) and does not reflect the singularities of (X, D). On the other hand, the adjoint ideal sheaf adj(X, ∆) of a boundary ∆ (i.e., ∆ = i d i ∆ i is an R-divisor with 0 ≤ d i ≤ 1) on X is a variant of the multiplier ideal sheaf J (X, ∆), and it encodes much information on the singularities of (X, ∆) even when ∆ is a Cartier integral divisor. Ein-Lazarsfeld [6] and Debarre-Hacon [4] used the adjoint ideal sheaf to study the singularities of ample divisors of low degree on abelian varieties. Kawakita [16] used the adjoint ideal sheaf to prove inversion of adjunction on log canonicity. The purpose of this paper is to give an interpretation of the adjoint ideal sheaf via a variant of tight closure.Tight closure is an operation defined on ideals or modules in positive characteristic. It was introduced by Hochster-Huneke [13] in the 1980s. The notions of F-regular rings and F-rational rings are defined via tight closure, and they turned out to correspond to log terminal and rational singularities, respectively ([7], [10], [21], [23]). This result is generalized to the correspondence of the test ideal and the multiplier ideal of the trivial divisor ([8], [24]). Here, the test ideal τ (R) of a Noetherian local ring (R, m) of prime characteristic p is the annihilator ideal of the tight closure 0 * E R (R/m) of the zero submodule in the injective hull E R (R/m) of the residue field R/m of R, and it plays a central role in the theory of tight closure. Since we can enjoy the usefulness of multiplier ideals only when they are associated to various ideals or divisors, Hara- In this paper, we introduce another generalization of tight closure associated to any given boundary, called divisorial tight closure, and investigate its properties. Let (R, m) be a Noetherian normal local ring of characteristic p > 0 and ∆ be a boundary on X := Spec R. Then the divisorial ∆-tight closure I div * ∆ of an ideal I ⊆ R is the ideal consisting of all elements x ∈ R for which there exists c ∈ R not in any minimal prime ideal of H 0 (X, O X (−⌊∆⌋)) such thatfor all large q = p e , where I [q] is the ideal generated by the q-th powers of elements of I. If N ⊆ M are R-modules, then the divisorial ∆-tig...

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“…Kawakita [16] used this ideal sheaf to prove inversion of adjunction on log canonicity and Shokurov [22] also used it implicitly.…”

Section: Introductionmentioning

confidence: 99%

A characteristic p analogue of plt singularities and adjoint ideals

Takagi

2007

Math. Z.

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“…Let Υ be such as in Lemma 2.5 and let K S + Θ be a non-klt 1, 2, 3, 4, or 6-complement of K S + Diff S . Using that the coefficients of Diff S are standard [Sh,Prop. 3.9] it is easy to see that Θ ≥ Diff S and Θ ≥ C [P3,Sect.…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Then Γ ≃ P 1 . There is an n-complement ∆ + = δ + i ∆ i of K S + ∆ near ϕ −1 (ô) for n ∈ {1, 2, 3, 4, 6} (see [Sh,Th. 5.6], [Ut,Cor.…”

Section: Consider the Case When G(s)mentioning

confidence: 99%

On the classification of Mori contractions: the case of an elliptic curve

Prokhorov¹

2001

Izv. Math.

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“…However, it turns out to be a very natural and useful condition arising in many questions. See [Shokurov 1992] and [Kollár 1994] for more detailed discussions of this. Thus the general conjecture is formulated as follows.…”

Section: Introductionmentioning

confidence: 99%