2. Other and Related Problems
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Problem 2.1.
[Takagi] Investigate the subadditivity of (Hacon-de Fernex) multiplier ideals in the non-\Q-Gorenstein case. -
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? -
Direct summand conjecture
Problem 2.35.
[Hochster] Prove/disprove the Direct Summand Conjecture. -
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}? -
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? -
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? -
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.