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Harbourne–Huneke containment conjecture

For every integer N≥2N\ge 2 and every integer s≥1s\ge 1, let I⊆k[x0,…,xN]I\subseteq k[x_0,\ldots,x_N] be the homogeneous defining ideal of a general set of ss points in PN\mathbb{P}^N over an algebraically closed field, and let m=(x0,…,xN)\mathfrak{m}=(x_0,\ldots,x_N). Then, for every integer r≥1r\ge 1, one has I(Nr)⊆mr(N−1)IrI^{(Nr)}\subseteq \mathfrak{m}^{r(N-1)}I^r.

References

Primary source

arXiv

Additional references

algebraic-geometrycommutative-algebra
Posted · edited

Progress summary

Refreshed
Claimed progress

A new preprint claims the large-exponent restriction can be removed for general point configurations, but it does not settle arbitrary configurations.

The Harbourne–Huneke conjecture asks for symbolic-power containments beyond the previously known regime where the exponent is sufficiently large. The relevant distinction is between general point configurations, which form a dense open family, and arbitrary ideals or configurations.

Known results

  • Earlier work established stable containments for sufficiently large rr for general points in PN\mathbb{P}^N, with a threshold depending on the number of points and NN.
  • Later work extended the stable result to every number of general points, still only for r>r(s,N)r>r(s,N).
  • Special configurations, including complements of Steiner configurations, also satisfy stable variants.
  • The corresponding statements can fail for some arbitrary configurations, including star configurations.

October 2026 claimed advance

Grzegorz Malara's preprint From Demailly's Inequality to Harbourne--Huneke Containments for General Points claims that an arbitrary-finite-set Demailly theorem combined with a strict generic Waldschmidt estimate removes the former large-rr threshold for a dense open family of general points in every projective dimension. The claim is not independently verified and does not cover all ideals or configurations.

Current status (as of October 2026): Stable containments for general points are established, while the claimed threshold-free generic result remains unverified and the statement for arbitrary configurations remains open.

Sources

Solutions 0

No solutions have been posted yet.