Reducedness of Hilb^8(S) for surfaces with ADE singularities

The problem asks whether $\operatorname{Hilb}^8(S)$ is reduced for surfaces $S$ with ADE singularities, extending the established range $n\leq 7$.

Known results

• Reducedness holds for $n\leq 7$ for surfaces with canonical singularities, hence for ADE surfaces (2026 source).
• For rational double points, $\operatorname{Hilb}^8(X)$ is reducible, and likewise for $d\geq 8$; this concerns non-smoothable subschemes, not nonreducedness (2017).
• The reduced underlying scheme $\operatorname{Hilb}^{[n]}(\mathbb{A}^2/\Gamma)_{\mathrm{red}}$ is irreducible, normal, and symplectic, but this does not establish reducedness of the full scheme.

July 2026 update

A 2026 arXiv paper explicitly states that reducedness in arbitrary degree remains unproved and gives no resolution for $n=8$. The scan found no proof, counterexample, verification, or AI-attributed claim addressing this case.

Current status (as of August 2026): reducedness is settled for $n\leq 7$, while the case $n=8$ and arbitrary $n$ remain open.