F-thresholds of hypersurfaces

By Manuel Blickle, Mircea Mustactva, Karen E. Smith. 2007

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′) for every λ ′ 0 that is F-finite, i.e. such that the Frobenius morphism F : R − → R is finite. If a is an ideal in R and if λ is a nonnegative real number, then the corresponding test ideal is denoted by τ(a � ). In this context we say that λ is an F-jumping exponent (or an F-threshold) if τ(a � ) 6 τ(a � ′ ) for every λ ′ < λ. The following is our main result about F-jumping exponents in positive characteristic. Theorem 1.1. If R is an F-finite regular ring, and if a = (f) is a principal ideal, then the F-jumping exponents of a are rational, and they form a discrete set. The discreteness and the rationality of F-jumping numbers has been proved in [BMS] for every ideal when the ring R is essentially of finite type over an F-finite field. We mention also that for R = k[[x, y]], with k a finite field, the above result has been proved in [Ha] using a completely different approach. We stress that the difficulty in attacking this result does notcome from the fact that there is no available resolution of singularities in positive characteristic. Even

Subfields: Algebra and Number Theory, Geometry and Topology

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Citations in the corpus

F-Thresholds, tight closure, integral closure, and multiplicity bounds (2007, methodology): “A positive answer was given in [ BMS2 ] and [BMS1] for a regular F-finite ring, essentially of finite type over a field, and for every regular F-finite ring, if the ideal a is principal.”
Discreteness and rationality of F-jumping numbers on singular varieties (2009, methodology): “normal scheme having a canonical divisor K X such that KX +∆ is Q-Cartier with index not divisible by p, and a is principal. In the case that X is regular and F-ﬁnite this result has been obtained in [BMS09] using D-module techniques and in [Bli08] using variants of a fundamental theorem of Hartshor”
Discreteness and rationality of F-thresholds (2006, methodology): “It is however used successfully in our preprint [BMS] to derive rationality and discreteness of F–thresholds of principal ideals in rings of formal power series, for example; thereby generalizing the 2 variable case treated in [HM].”
IMPANGA lecture notes on log canonical thresholds (2011, background): “ollowing result from [BMS2] gives the analogue for the rationality and the discreteness of the jumping numbers of the multiplier ideals of a given ideal. For extensions to various other settings, see [BMS1], [KLZ] and [BSTZ]. Theorem 2.17. If ais a nonzero ideal in R, then the set of F-jumping numbe”
p −1-Linear Maps in Algebra and Geometry (2012, result): “ls and multiplier ideals [HY03, Tak04] it is natural to expect that the set of jumping numbers for the test ideal is discrete. In the case that Xis smooth this was shown to be the case in [BMS08] and [BMS09]. The singular case was obtained in [BSTZ10], see also [Har06, KLZ09, ST08, TT08, STZ12, AMBZ”