Loading Web-Font TeX/Math/Italic
| Register
\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

2. Other and Related Problems

    1. Problem 2.05.

      [de Fernex] 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?
        • Problem 2.1.

          [Takagi] Investigate the subadditivity of (Hacon-de Fernex) multiplier ideals in the non-\Q-Gorenstein case.
            1. Remark. This is closely related to problem 22.05.
                • Problem 2.15.

                  [Ishii] Define a “test ideal" that relates to the Mather discrepancy multiplier ideal, in the non-\mathbb{Q}-Gorenstein case.
                    • Problem 2.2.

                      [Patakfalvi] Investigate lifting sections (for cohomology) using test ideals.
                        • Problem 2.25.

                          [Mustaţă] 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?
                            • Problem 2.3.

                              [Hochster] Realize effective computations of test ideals.
                                • Direct summand conjecture

                                  Problem 2.35.

                                  [Hochster] Prove/disprove the Direct Summand Conjecture.
                                    • Problem 2.4.

                                      [Hochster] Extend tight closure to mixed characteristic.
                                        • Problem 2.45.

                                          [Schwede] 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}?
                                              Answered positively by Huneke-Watanabe at the AIM workshop.
                                            • Problem 2.5.

                                              [Lyubeznik] 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?
                                                • Problem 2.55.

                                                  [Singh] Do determinantal rings have finite F-representation type?
                                                      The 2 \times 2 minors of a generic matrix has been done by Smith and Van den Bergh; try 3 \times 3 minors and higher.
                                                    • Problem 2.6.

                                                      [Brenner] For p \geq 11, does \mathbb{F}_p[x,y,z]/(x^2+y^3+z^7) have finite F-representation type?
                                                        • Problem 2.65.

                                                          [Schwede] 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?
                                                              This is known in the Cohen-Macaulay case. An analogous result is known to hold for Du Bois singularities in characteristic zero, in generality.
                                                            • Problem 2.7.

                                                              [Blickle] 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?
                                                                • Problem 2.75.

                                                                  [Mustaţă] Identify a possible positive characteristic analog of minimal log discrepancy.
                                                                    • Problem 2.8.

                                                                      [Schwede/Patakfalvi] Investigate possible Bertini theorems for F-singularities.
                                                                        • Problem 2.85.

                                                                          [Joshi] Investigate the applications of the topics of the conference (e.g., test ideals) to projective geometry.

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