All-r shifted Harbourne–Huneke containments in higher dimensions

Determine whether the method based on Demailly’s inequality and sharper finite interpolation data yields the shifted containment I^{(Nr-N+1)}\subseteq \mathfrak{m}^{(r-1)(N-1)}I^r for every r\ge1 for general points in projective space of higher dimension.

Background

The paper studies the strengthened Harbourne–Huneke containment for ideals of general points and proves that Demailly’s inequality, together with a strict generic Waldschmidt estimate, gives the unshifted containment simultaneously for all symbolic exponents on one dense Zariski-open family.

The authors then consider the stronger shifted containment I^{(Nr-N+1)}\subseteq \mathfrak{m}^{(r-1)(N-1)}I^r. They explain that their method provides effective stable ranges for this statement, but that a positive finite-level margin does not automatically handle the finitely many small values of r. The all-r shifted result is known in certain cases, including projective 3-space, but remains unresolved in higher dimensions under the method discussed.