Discreteness and rationality of F-thresholds

Discreteness and rationality of $F$-jumping numbers on singular varieties

We prove that the $F$-jumping numbers of the test ideal $τ(X; Δ, \ba^t)$ are discrete and rational under the assumptions that $X$ is a normal and $F$-finite variety over a field of positive characteristic $p$, $K_X+Δ$ is $\bQ$-Cartier of index not divisible $p$, and either $X$ is essentially of finite type over a field or the sheaf of ideals $\ba$ is locally principal. This is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero.

On the discreteness and rationality of F-jumping coefficients

This paper studies the jumping coefficients of principal ideals of regular local rings. Recently M. Blickle, M. Mustata and K. Smith showed that, when $R$ is of essentially finite type over a field and $F$-finite, bounded intervals contain finitely many jumping coefficients and that those are rational. In a later paper they extended these results to principal ideals of $F$-finite complete regular local rings. The aim of this paper is to extend these results on the discreteness and rationality of jumping coefficients to principal ideals of arbitrary (i.e. not necessarily $F$-finite) excellent r…

Test ideals via a single alteration and discreteness and rationality of $F$-jumping numbers

Suppose $(X, Δ)$ is a log-$\bQ$-Gorenstein pair. Recent work of M. Blickle and the first two authors gives a uniform description of the multiplier ideal $\mJ(X;Δ)$ (in characteristic zero) and the test ideal $τ(X;Δ)$ (in characteristic $p > 0$) via regular alterations. While in general the alteration required depends heavily on $Δ$, for a fixed Cartier divisor $D$ on $X$ it is straightforward to find a single alteration (e.g. a log resolution) computing $\mJ(X; Δ+ λD)$ for all $λ\geq 0$. In this paper, we show the analogous statement in positive characteristic: there exists a single regular al…

关于 $F$-阈值相对于吹脱纤维的性质以及 $F$-理性

On the behavior of $F$-thresholds with respect to the fibers of blow-ups and $F$-rationality

本文研究了 $F$-阈值在吹脱纤维下的行为，并证明了格雷德环的 $F$-理性性可传递到局部环。

完美oid纯度的判别准则与阈值的有理性

A Criterion for Perfectoid Purity and the Rationality of Thresholds

本文提出一个判断超曲面完美oid纯度的准则，证明阈值为有理数，并揭示了大素数下Fermat型Calabi-Yau超曲面锥的完美oid纯度例子。

On Fujita's spectrum conjecture

We prove Fujita's spectrum conjecture on the discreteness of pseudo-effective thresholds for polarized varieties.