The minimal log discrepancies on a smooth surface in positive characteristic

Ishii, Shihoko

doi:10.1007/s00209-020-02514-8

Cited by 6 publications

(11 citation statements)

References 9 publications

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“…In the proof of Theorem 1.4 in [Is2], Ishii proves that there is a regular system of parameters x, y of O A,0 and a monomial R-ideal a * on A 2 k with same exponents as a such that…”

Section: Minimal Log Discrepancymentioning

confidence: 99%

“…In this paper, we will use a completely different approach to give a smaller bound which belongs to O( 1 γ 2 ) as γ → 0. Our idea comes from Ishii [Is2]. In the paper, Ishii proves that MN conjecture holds for any smooth surface A of arbitrary characteristic and she points out that in surfaces case the upper bound in the conjecture can be calculated by using toric geometry.…”

Section: Introductionmentioning

confidence: 99%

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Upper bound of discrepancies of divisors computing minimal log discrepancies on surfaces

Chen¹

2020

Preprint

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“…In the proof of Theorem 1.4 in [Is2], Ishii proves that there is a regular system of parameters x, y of O A,0 and a monomial R-ideal a * on A 2 k with same exponents as a such that…”

Section: Minimal Log Discrepancymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Upper bound of discrepancies of divisors computing minimal log discrepancies on surfaces

Chen¹

2020

Preprint

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“…Here, we note that the second horizontal line is a sequence of morphisms of smooth surfaces over an algebraically closed field K. So, we can apply the discussion for surfaces in [9] and [6].…”

Section: Take a General Element Fmentioning

confidence: 99%

“…Actually, in the paper [9]and [6], the main theorem is not stated in this form, but its proof shows Theorem 1.1. The paper [9] is for char k = 0, and the paper [6] is for char k = p > 0 and the main statements of the both papers are in the following form:…”

Section: Introductionmentioning

confidence: 99%

A bound of the number of weighted blow-ups to compute the minimal log discrepancy for smooth 3-folds

Ishii

2021

Preprint

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We study a pair consisting of a smooth 3-fold defined over an algebraically closed field and a "general" real ideal. We show that the minimal log discrepancy ("mld" for short) of every such a pair is computed by a prime divisor obtained by at most two weighted blow-ups. This bound is regarded as a weighted blow-up version of Mustaţǎ-Nakamura's conjecture. We also show that if the mld of such a pair is not less than 1, then it is computed by at most one weighted blow-up. As a consequence, ACC of mld holds for such pairs. Corollary 1.10 (Corollary 5.9). Assume N = 3. Let E be a prime divisor over A computing mld(0; A, a) for a pair (A, a). Assume that the center C ⊂ A 1 of E on the first blow-up A 1 −→ A is a curve of degree ≥ 2 in the exceptional divisor E 1 ≃ P 2 . Then a "standard blow-up" at C gives a divisor computing mld(0; A, a).Note that in this case the first blow-up is also a standard blow-up. Example 3.2 is just in this case. In Section 5, we show a more general corollary. On the other hand, if we restrict to the case mld ≥ 1, then we have the following:Theorem 1.11. Assume N = 3. Then, for every general pair (A, a) with a and mld(0; A, a) ≥ 1 , the minimal log discrepancy is computed by a prime divisor obtained by one weighted blow-up.Corollary 1.12. Assume N = 3. In {(A, a) | mld(0; A, a) ≥ 1 with general a} Mustaţǎ-Nakamura Conjecture holds and also ACC Conjecture holds for char k ≥ 0.

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“…Although the study on minimal log discrepancies was traditionally considered over C, recently there has been some studies on the structure of minimal log discrepancies over fields of arbitrary characteristics (cf. [7,17,36]). In this paper, the results hold over fields of arbitrary characteristics.…”

Section: Theorem 1•2 and Theorem 1•4 Follow From The Following Classi...mentioning

confidence: 99%

Divisors computing minimal log discrepancies on lc surfaces

Liu¹,

Xie²

2023

Math. Proc. Camb. Phil. Soc.

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Let $(X\ni x,B)$ be an lc surface germ. If $X\ni x$ is klt, we show that there exists a divisor computing the minimal log discrepancy of $(X\ni x,B)$ that is a Kollár component of $X\ni x$ . If $B\not=0$ or $X\ni x$ is not Du Val, we show that any divisor computing the minimal log discrepancy of $(X\ni x,B)$ is a potential lc place of $X\ni x$ . This extends a result of Blum and Kawakita who independently showed that any divisor computing the minimal log discrepancy on a smooth surface is a potential lc place.

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The minimal log discrepancies on a smooth surface in positive characteristic

Cited by 6 publications

References 9 publications

Upper bound of discrepancies of divisors computing minimal log discrepancies on surfaces

Upper bound of discrepancies of divisors computing minimal log discrepancies on surfaces

A bound of the number of weighted blow-ups to compute the minimal log discrepancy for smooth 3-folds

Divisors computing minimal log discrepancies on lc surfaces

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