Stable Harbourne–Huneke containment conjecture

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. Stable Harbourne–Huneke containment conjecture

   Let $I\subseteq{\mathbbm k}[{\mathbb P}^N_{\mathbbm k}]$ be a homogeneous radical ideal of big height $h$. Then there is a constant $r(I)\geqslant1$, depending on $I$, such that for every $r\geqslant r(I)$,

   $$I^{(hr)}\subseteq{\mathfrak m}^{r(h-1)}I^r$$

   and

   $$I^{(hr-h+1)}\subseteq{\mathfrak m}^{(r-1)(h-1)}I^r.$$

   Stable Harbourne–Huneke containment conjecture. Both containments hold for all sufficiently large $r$ as specified above. The stable containment is presented as implying Chudnovsky's conjecture; the paper's abstract claims the conjectures for all numbers of general points, while the supplied status remains unknown.

   source: Sankhaneel Bisui and Thái Thành Nguyên, “Chudnovsky's Conjecture and the stable Harbourne-Huneke containment for general points”, arXiv:2112.15260 (2022).