
## 1. Characteristic $p>0$ invariants

1. #### Problem 1.05.

[Katzman] Given an ideal $I$ of $\Z[x]$, the HSL numbers are the “indices of nilpotency" of the action of Frobenius on $H^\text{top}_{(\underline{x})}\left( \Z_p[\underline{x}]/I_p \right)$, for primes $p$. Is the limsup of the HSL numbers 1?
• #### Problem 1.1.

[McCullough] Realize effective computations of numerical $F$-invariants.
• #### Problem 1.15.

[Hernández] Do Howald’s nondegeneracy conditions with respect to the Newton polyhedron on the term ideal provide conditions under which $\operatorname{fpt}(\text{polynomial}) = \operatorname{fpt}(\text{associated term ideal})$?
• #### Problem 1.2.

[Miller/Mustaţă] For a graded sequence of ideals $I_m$ of a ring $R$, does $\lim \limits_{m \to \infty} m \cdot \operatorname{fpt}( I_m)$ exist?
1. Remark. This is known for log canonical thresholds.
• #### Problem 1.25.

[Li] Investigate the existence and rationality of $F$-thresholds.
1. Remark. Existence is known for a ring that is F-pure on the punctured spectrum.
• #### Problem 1.3.

[Miller] Investigate the discreteness and rationality of $F$-jumping numbers in the non-$\Q$-Gorenstein case.
• #### Problem 1.35.

[Watanabe, Huneke] Fix $d$. Does there exist a sequence of singular algebras $\{ R_n \}$ over a fixed finite field of characteristic $p>0$, and of dimension $d$, such that the $e_{HK}\left(R_n\right)$ descend to some number $\alpha_d$ from above?
• #### Problem 1.4.

[Watanabe] Fix $d$. What is the maximum value of the $F$-pure threshold of $\mathfrak{m}$ of a non-$\Q$-Gorenstein local ring $(R, \mathfrak{m})$ of dimension $d$?
Such a maximum value is known to be $d-1$ for non-regular $\mathbb{Q}$-Gorenstein rings.
• #### Problem 1.45.

[Singh] Given a determinantal ring $R$, compute $e_{HK}(R)$ and $s(R)$.
The 2 $\times$ 2 minors of a generic matrix is done by Watanabe and Yoshida; try 3 $\times$ 3 minors and higher.
• #### Problem 1.5.

[Enescu] Investigate the upper semicontinuity of the map $\mathfrak{p} \in \operatorname{Spec} R \mapsto e_{HK}\left(R_\mathfrak{p} \right)$.
• #### Problem 1.55.

[Enescu] Investigate the lower semicontinuity of the map $\mathfrak{p} \in \operatorname{Spec} R \mapsto s\left(R_\mathfrak{p}\right)$.
• #### Problem 1.6.

[Tucker] If $a_e$ denote of $F$-splitting numbers of a strongly $F$-regular local ring $R$ (i.e., $R^{1/p^e} \cong R^{a_e} \oplus M$ as $R$-modules, and $M$ contains no free $R$-module summands), then is $a_e = s(R) p^{ed} + C p^{e(d-1)} + O \left(p^{e(d-2)}\right)$ for some constant $C$?
1. Remark. In the Gorenstein case, it is known and C=0.
• #### Problem 1.65.

[Tucker] For $(R, \mathfrak{m})$ a local, normal domain of characteristic $p>0$ and dimension $d$, let $\mu_e$ denote the minimal number of generators of $\operatorname{Hom}_R (R^{1/p^e}, R)$ as an $R$-module. Does $\lim \limits_{e \to \infty} \frac{\mu_e}{p^{ed}}$ exist? If so, what is it?
1. Remark. In the $\Q$-Gorenstein case, it exists and equals $e_{HK}{R}$.
• #### Problem 1.7.

[Aberbach] For $(R, \mathfrak{m})$ a local, normal domain of characteristic $p>0$, let $x$ be a minimal generator of $\mathfrak{m}$, and let $v$ denote a $t^\text{th}$ root of $x$. Relate $e_{HK}(R)$ and $e_{HK}\left(R[v]\right)$.

Cite this as: AimPL: Test ideals and multiplier ideals, available at http://aimpl.org/testandmultiplierideals.