Validity of the asymptotic results for every Fuss parameter

Prove that Theorems B and D hold for every integer $m\ge1$, equivalently establish the vanishing of higher cohomology and the required base-change isomorphism for the twists of the symbolic punctual fiber in all positive degrees.

Background

The principal results are established only for sufficiently large Fuss parameter m, with the threshold depending non-effectively on n. The relevant geometric obstruction is possible higher cohomology or failure of base change for the polarized symbolic punctual fiber.

A Cohen–Macaulay property of the entire punctual section ring, together with an appropriate bound on its a-invariant, is identified as a possible route to proving all positive twists simultaneously.

References

Do Theorems B and D hold for every $m\ge1$? By \cref{thm:character} this reduces to $H{>0}(_n{\mathrm{symb},(m))=0$ and base change in the finitely many low degrees.

— Asymptotic $q,t$-Fuss--Catalan numbers for type $B$  (2609.11691 - Oblomkov, 10 Sep 2026) in Section 16, subsection “Open problems,” Question “All twists”

Its bijectivity for all $n$, equivalently $H{>0}$-vanishing at $m=1$, would give the exact $q,t$-Fuss--Catalan character at the classical rung, and it would prove Haiman's conjecture Conj.~7.2.5 on the determinantal part of the diagonal coinvariants of $W(B_n)$.

— Asymptotic $q,t$-Fuss--Catalan numbers for type $B$  (2609.11691 - Oblomkov, 10 Sep 2026) in Section 16, subsection “Open problems,” Question “The first rung”