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1. Characteristic p>0 invariants


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      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?


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          Problem 1.1.

          [McCullough] Realize effective computations of numerical F-invariants.


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              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})?


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                  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.


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                          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.


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                                  Problem 1.3.

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


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                                      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?


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                                          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.


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                                              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.


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                                                  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).


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                                                      Problem 1.55.

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


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                                                          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.


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                                                                  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}.


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                                                                          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.