Let mathbbm k be a field, let P^N_mathbbm k be projective space, and let I⊆mathbbm k[ P^N_mathbbm k] be a homogeneous radical ideal of big height h. Let mathfrak m denote the graded irrelevant ideal. The stable Harbourne–Huneke containment conjecture asserts that, for any r≫0, I^(hr)⊆mathfrak m^r(h-1)I^r and I^(hr-h+1)⊆mathfrak m^(r-1)(h-1)I^r. The stable formulation implies Chudnovsky's conjecture. The paper proves the two containments for general sets of sufficiently many points in characteristic zero, while the general radical-ideal statement remains open.

Stable Harbourne–Huneke containment conjecture

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Let $\mathbb k$ be a field, let $\mathbb P^N_{\mathbb k}$ be projective space, and let $I\subseteq\mathbb k[\mathbb P^N_{\mathbb k}]$ be a homogeneous radical ideal of big height $h$ . Let $\mathfrak m$ denote the graded irrelevant ideal. The stable Harbourne–Huneke containment conjecture asserts that, for any $r\gg0$ ,

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

and

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

The stable formulation implies Chudnovsky's conjecture. The paper proves the two containments for general sets of sufficiently many points in characteristic zero, while the general radical-ideal statement remains open.

Equivalent formulations 1Other wordings

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

Stable Harbourne–Huneke containment conjecture
Let $I\subseteq{\mathbb k}[{\mathbb P}^N_{\mathbb k}]$ 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).

References

Primary source

Sankhaneel Bisui, Eloísa Grifo, Huy Tài Hà and Thái Thành Nguyên, “Chudnovsky's Conjecture and the stable Harbourne-Huneke containment”, arXiv:2004.11213 (2021).

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