General Hyperplane Sections of Threefolds in Positive Characteristic

Sato, Kenta; Takagi, Shunsuke

doi:10.1017/s1474748018000166

Cited by 12 publications

(13 citation statements)

References 40 publications

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“…Consider the integer ν := ν f (p e ) as in Lemma 2. 15. We note that it follows from Lemma 2.15 that we have ν ≡ t (e) ( mod p) and hence, ν + 1 is not divisible by p. By the definition of ν f (p e ), we have…”

Section: Construction Of Counterexamplesmentioning

confidence: 91%

On accumulation points of $F$-pure thresholds on regular local rings

Sato¹

2020

Preprint

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Blickle, Mustat ¸ȃ and Smith proposed two conjectures on the limits of F -pure thresholds. One conjecture asks whether or not the limit of a sequence of F -pure thresholds of principal ideals on regular local rings of fixed dimension can be written as an F -pure thresshold in lower dimension. Another conjecture predicts that any F -pure threshold of a formal power series can be written as the F -pure threshold of a polynomial. In this paper, we prove that the first conjecture has a counterexample but a weaker statement still holds. We also give a partial affirmative answer to the second conjecture.

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Section: Construction Of Counterexamplesmentioning

confidence: 91%

On accumulation points of $F$-pure thresholds on regular local rings

Sato¹

2020

Preprint

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“…This completes the proof of Claim 2. Now let the defining ideal of X n (x) in A n (x) be I n for n ∈ N. Then, for n = (α + 1)l − 1, we have (14) htI (α+1)m−α 2 ≤ htJ m + (m − α)N, which will give the required inequality. Actually, to see this, remind us the definition of s m (X, x) and s m (Y, 0) to obtain:…”

Section: Some Affirmative Casesmentioning

confidence: 99%

“…One good example is Sato-Takagi's result ( [14]) that for a quasi-projective 3-fold X with canonical singularities over an algebraically closed field of characteristic p > 0, a general hyperplane section of X has also canonical singularities. They proved this by making use of a result about MJ-canonical singularities proved in [9].…”

Section: Conjecture 12 (C Dδmentioning

confidence: 99%

“…14) htI (α+1)m−α 2 ≤ htJ m + (m − α)N, which will give the required inequality. Actually, to see this, remind us the definition of s m (X, x) and s m (Y, 0) to obtain:s m (Y, 0) = (m + 1)d − mN + htJ m = −cm + d + htJ m , s (α+1)m−α 2 (X, x) = −c((α + 1)m − α 2 ) + d + htI (α+1)m−α 2 .Substituting these into(14) and by noting that N = α • c, we finally obtain…”

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confidence: 92%

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Finite determination conjecture for Mather–Jacobian minimal log discrepancies and its applications

Ishii

2018

European Journal of Mathematics

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In this paper we study singularities in arbitrary characteristic. We propose Finite Determination Conjecture for Mather-Jacobian minimal log discrepancies in terms of jet schemes of a singularity. The conjecture is equivalent to the boundedness of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy. We also show that this conjecture yields some basic properties of singularities; eg., openness of Mather-Jacobian (log) canonical singularities, stability of these singularities under small deformations and lower semi-continuity of Mather-Jacobian minimal log discrepancies, which are already known in characteristic 0 and open for positive characteristic case. We show some evidences of the conjecture: for example, for non-degenerate hypersurface of any dimension in arbitrary characteristic and 2-dimensional singularities in characteristic not 2. We also give a bound of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy.Mathematical Subject Classification: 14B05,14E18, 14B07

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“…Throughout this paper, all rings will be assumed to be -finite and of characteristic . If is an -finite Noetherian normal ring, then is excellent [Kun76] and has a canonical divisor (see, for example, [ST18, p. 4]).…”

Section: Preliminariesmentioning

confidence: 99%

Ascending chain condition for -pure thresholds on a fixed strongly -regular germ

Sato¹

2019

Compositio Math.

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In this paper, we prove that the set of all $F$-pure thresholds on a fixed germ of a strongly $F$-regular pair satisfies the ascending chain condition. As a corollary, we verify the ascending chain condition for the set of all $F$-pure thresholds on smooth varieties or, more generally, on varieties with tame quotient singularities, which is an affirmative answer to a conjecture given by Blickle, Mustaţǎ and Smith.

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General Hyperplane Sections of Threefolds in Positive Characteristic

Cited by 12 publications

References 40 publications

On accumulation points of $F$-pure thresholds on regular local rings

On accumulation points of $F$-pure thresholds on regular local rings

Finite determination conjecture for Mather–Jacobian minimal log discrepancies and its applications

Ascending chain condition for -pure thresholds on a fixed strongly -regular germ

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