MathDB

Briançon–Iarrobino Conjecture and Conjecture B on maximal singularity of threefold Hilbert schemes

For a smooth threefold XX and an integer ℓ≥1\ell\geq 1, characterize the points [Z]∈Hilb⁡ℓ(X)[Z]\in \operatorname{Hilb}^{\ell}(X) having maximal singularity, namely those satisfying dim⁡T[Z]Hilb⁡ℓ(X)=max⁡[W]∈Hilb⁡ℓ(X)dim⁡T[W]Hilb⁡ℓ(X)\dim T_{[Z]}\operatorname{Hilb}^{\ell}(X)=\max_{[W]\in\operatorname{Hilb}^{\ell}(X)}\dim T_{[W]}\operatorname{Hilb}^{\ell}(X). The Briançon–Iarrobino conjectures propose numerical necessary conditions for such points and, in particular, assert the conjectured maximal-singularity behavior in the tetrahedral-degree cases.

References

Primary source

arXiv

Additional references

algebraic-geometryhilbert-schemessingularity-theory
Posted

Progress summary

Refreshed
Claimed progress

A new unrefereed manuscript provides substantial partial evidence for the conjectures, but the general problem remains open.

The Briançon–Iarrobino conjectures concern which points of threefold Hilbert schemes are maximally singular; the principal conjecture dates to 1978. Conjecture B proposes a necessary numerical condition for such maximally singular points.

Known results

  • An older preprint gives partial tangent-space results and examples for the first and second Briançon–Iarrobino conjectures, but the retrieved summary does not specify its exact range.
  • A 2025 preprint proposes monotonicity of tangent-space dimension and reports computational confirmation in dimension 33 for 10≤ℓ≤3510\leq \ell\leq 35.
  • Another preprint gives conjectural necessary and sufficient criteria, including computational examples, but also a length-1616 example showing its proposed necessary condition is not sufficient.

September 3, 2026 partial advance

A new manuscript proves tangent-space upper and lower bounds establishing Conjecture B’s proposed necessary condition for a broad degree range and implying the 1978 conjecture in a specific tetrahedral case. On August 29, 2025, Owen Mackenzie and Fatemeh Rezaee had claimed the three-dimensional tetrahedral case, but that preprint remains unrefereed; no independent verification or counterexample was found.

Current status (as of September 2026): Specific tetrahedral three-dimensional cases are claimed in unrefereed preprints, and a broad necessary condition has been claimed, but the general Briançon–Iarrobino conjecture and Conjecture B remain open.

Sources

Solutions 0

No solutions have been posted yet.