Discreteness and rationality of F-thresholds

Blickle, Manuel; Mustaţă, Mircea; Smith, Karen E.

doi:10.1307/mmj/1220879396

Cited by 127 publications

(223 citation statements)

References 13 publications

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“…Recall first from [BMS,Proposition 3.6] that the denominators of the F -jumping numbers of f in the above set are bounded in terms of d, n and the characteristic p; in particular, the bound is independent of the field k. Therefore we have 0 < a 1 < · · · < a m (d,n,p) ≤ 1 such that for every nonzero f of degree ≤ d and with f (0) = 0, we have fpt 0 (f ) = a i for some i. From now on we may assume that c = a j for some j.…”

Section: Is Algebraically Closed (From Now On We Simply Denote This Smentioning

confidence: 99%

$F$-thresholds of hypersurfaces

Blickle

Mustaţă

Smith

2009

Trans. Amer. Math. Soc.

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Section: Is Algebraically Closed (From Now On We Simply Denote This Smentioning

confidence: 99%

$F$-thresholds of hypersurfaces

Blickle

Mustaţă

Smith

2009

Trans. Amer. Math. Soc.

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“…Where the Fjumping exponents of the test ideals of g are the pendant to the jumping coefficients of the multiplier ideals. These are rational numbers [6,Theorem 3.1].…”

Section: Positive Characteristicmentioning

confidence: 99%

On a theory of the b-function in positive characteristic

Bitoun

2018

Sel. Math. New Ser.

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We present a theory of the b-function (or Bernstein-Sato polynomial) in positive characteristic. Let f be a non-constant polynomial with coefficients in a perfect field k of characteristic p > 0. Its b-function b f is defined to be an ideal of the algebra of continuous k-valued functions on Z p . The zero-locus of the b-function is thus naturally interpreted as a subset of Z p , which we call the set of roots of b f . We prove that b f has finitely many roots and that they are negative rational numbers. Our construction builds on an earlier work of Mustaţȃ and is in terms of D-modules, where D is the ring of Grothendieck differential operators. We use the Frobenius to obtain finiteness properties of b f and relate it to the test ideals of f.

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“…So as to give an idea where the difficulties lie, recall that the classical construction of the test ideal τ (R, a t ) [HY03,Tak04,BMS08] is based upon a number of manipulations of ideals involving the Frobenius morphism or pth power map. For example, the p e th Frobenius or bracket power b [p e ] of an ideal b ⊆ R is the expansion under the e-th iterate of Frobenius and is generated by the p e th powers of elements of b.…”

Section: Introductionmentioning

confidence: 99%

Test ideals of non-principal ideals: Computations, jumping numbers, alterations and division theorems

Schwede

Tucker

2014

Journal de Mathématiques Pures et Appliquées

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Abstract. Given an ideal a ⊆ R in a (log) Q-Gorenstein F -finite ring of characteristic p > 0, we study and provide a new perspective on the test ideal τ (R, a t ) for a real number t > 0. Generalizing a number of known results from the principal case, we show how to effectively compute the test ideal and also describe τ (R, a t ) using (regular) alterations with a formula analogous to that of multiplier ideals in characteristic zero. We further prove that the F -jumping numbers of τ (R, a t ) as t varies are rational and have no limit points, including the important case where R is a formal power series ring. Additionally, we obtain a global division theorem for test ideals related to results of Ein and Lazarsfeld from characteristic zero, and also recover a new proof of Skoda's theorem for test ideals which directly mimics the proof for multiplier ideals.

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Discreteness and rationality of F-thresholds

Cited by 127 publications

References 13 publications

$F$-thresholds of hypersurfaces

$F$-thresholds of hypersurfaces

On a theory of the b-function in positive characteristic

Test ideals of non-principal ideals: Computations, jumping numbers, alterations and division theorems

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