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\begin{document}
\title{Test ideals and multiplier ideals}
\author{Edited by Emily Witt}

\urlstub{testandmultiplierideals}

\maketitle




\section{Characteristic $p>0$ invariants}





\begin{problemblock}

\begin{problem}[1.05] 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? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[1.1] Realize effective computations of numerical $F$-invariants. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[1.15] Do Howald's nondegeneracy conditions with respect to the Newton polyhedron on the term ideal provide conditions under which $\operatorname{fpt}(\text{polynomial}) = \operatorname{fpt}(\text{associated term ideal})$? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[1.2] 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?  \end{problem}




\begin{remark}This is known for log canonical thresholds.\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[1.25] Investigate the existence and rationality of $F$-thresholds. \end{problem}




\begin{remark}Existence is known for a ring that is F-pure on the punctured spectrum.\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[1.3] Investigate the discreteness and rationality of $F$-jumping numbers in the non-$\Q$-Gorenstein case. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[1.35] 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? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[1.4] 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 $d$? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[1.45] Given a determinantal ring $R$, compute $e_{HK}(R)$ and $s(R)$.  \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[1.5] Investigate the upper semicontinuity of the map $\mathfrak{p} \in \operatorname{Spec} R \mapsto e_{HK}\left(R_\mathfrak{p} \right)$. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[1.55] Investigate the lower semicontinuity of the map $\mathfrak{p} \in \operatorname{Spec} R \mapsto s\left(R_\mathfrak{p}\right)$.  \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[1.6] 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 $C$?  \end{problem}




\begin{remark}In the Gorenstein case, it is known and C=0.\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[1.65] 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?   \end{problem}




\begin{remark}In the $\Q$-Gorenstein case, it exists and equals $e_{HK}{R}$.\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[1.7] 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 $e_{HK}\left(R[v]\right)$. \end{problem}




\end{problemblock}



\section{Other and Related Problems}





\begin{problemblock}

\begin{problem}[2.05] Generalize the Hara-Yoshida Theorem on restriction of (Hacon-de Fernex) multiplier ideals to test ideals to the non-$\Q$-Gorenstein case.  Can this be done in the numerically Gorenstein setting?\label{generalizedHaraYoshida} \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.1] Investigate the subadditivity of (Hacon-de Fernex) multiplier ideals in the non-$\Q$-Gorenstein case.  \end{problem}




\begin{remark}This is closely related to problem \ref{generalizedHaraYoshida}.\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[2.15] Define a ``test ideal" that relates to the Mather discrepancy multiplier ideal, in the non-$\mathbb{Q}$-Gorenstein case.   \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.2] Investigate lifting sections (for cohomology) using test ideals. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.25] Given ideals $I_1, \ldots, I_n$ of a regular, $F$-finite ring $R$, consider the map sending an $r$-tuple of nonnegative real numbers $(\lambda_1, \ldots, \lambda_r)$ to $\tau\left( {I_1}^{\lambda_1} \cdot \ldots \cdot{I_n}^{\lambda_n}\right)$, the mixed test ideal.  If we bound all the $\lambda_i$ by some fixed $M$, does there exist a rational polyhedral decomposition such that the function described is constant on the interior of each region? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.3] Realize effective computations of test ideals. \end{problem}




\end{problemblock}



\begin{problemblock}
\name{Direct summand conjecture}


\begin{problem}[2.35] Prove/disprove the Direct Summand Conjecture. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.4] Extend tight closure to mixed characteristic. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.5] Given a smooth, projective variety $X$ over a number field,  does there exist a dense set of primes for which the action of Frobenius on the coherent cohomology $H^i(X_p, \mathcal{O}_{X_p})$ is not nilpotent?
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.45] Suppose that $R$ is a local, equidimensional, $F$-pure ring of dimension $d$ with embedding dimension $n$.  Is the multiplicity of $R$ at most $\binom{n}{d}$?   \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.55] Do determinantal rings have finite $F$-representation type? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.6] For $p \geq 11$, does $\mathbb{F}_p[x,y,z]/(x^2+y^3+z^7)$ have finite $F$-representation type? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.65] Suppose that $R$ is a local ring and $f \in R$ is a regular element such that $R/(f)$ is $F$-injective. Does this imply that $R$ is $F$-injective?  \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.7] Given a log pair $(X ,\Delta)$, we know that there exists a finite map $\phi: Y \to X$ such that $\operatorname{im}\left(\phi_{*} \mathcal{O}_Y \left( K_Y - \phi^*\left( K_X + \Delta \right)\right) \overset{\text{trace}}{\longrightarrow} \mathcal{O}_X \right) = \tau(X, \Delta)$.  Is this statement true if further decorated by $\mathfrak{a}^t$?  What about for alterations? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.75] Identify a possible positive characteristic analog of minimal log discrepancy. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.8] Investigate possible Bertini theorems for $F$-singularities. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[2.85] Investigate the applications of the topics of the conference (e.g., test ideals) to projective geometry. \end{problem}




\end{problemblock}





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\end{document}
