Reducedness and irreducibility conjecture for Weddle schemes

Less than 1 year old · traced to 2026

Let $\Sigma$ be the $d$ -Weddle scheme associated with a general set of $\binom{d+n}{n}$ points in $\mathbb{P}^{n+1}$ , as in the preceding theorem. Reducedness and irreducibility conjecture. The scheme $\Sigma$ is reduced and irreducible. This conjecture concerns the geometric structure of the determinantal Weddle scheme; the preceding theorem establishes its defining degree, but the supplied text gives no resolution of reducedness or irreducibility.

References

Primary source

Luca Chiantini, Łucja Farnik, Giuseppe Favacchio, Brian Harbourne, Juan Migliore, Tomasz Szemberg and Justyna Szpond, “Weddle schemes”, arXiv:2606.25060 (2026).

algebraic-geometry weddle-schemes

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

The conjecture remains publicly unproved, while an unverified submission claims it holds in dimensions two and higher but fails in dimension one.

A 2026 preprint formulates the conjecture that the Weddle scheme associated with general points is reduced and irreducible. It proves that the scheme is a hypersurface of degree $\binom{d+n}{n+1}$ , but reports no resolution of the conjecture.

Known results

For general points, the scheme is a hypersurface of degree $\binom{d+n}{n+1}$ (2026 preprint).
Special configurations can produce nonreduced or nonequidimensional schemes; these do not contradict the general-point conjecture.

Community submission (unverified), August 25, 2026

A submitted argument claims that the general $d$ -Weddle hypersurface is integral for $n\geq 2$ , while for $n=1$ it is reduced but reducible. If correct, this would prove the conjecture for $n\geq 2$ and disprove its irreducibility assertion for $n=1$ , but the argument is unverified.

Current status (as of August 2026): The hypersurface and degree are settled, but reducedness and irreducibility remain open in the public record; the dimension-dependent submission is unverified.

Sources

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For $n\geq2$ , the $d$ -Weddle hypersurface associated with $\binom{d+n}{n}$ general points in $\mathbb{P}^{n+1}$ is integral. If $n=1$ , the general hypersurface is reduced but not irreducible.