Limiting Betti distributions of Hilbert schemes on $n$ points

TL;DR

This paper demonstrates that the limiting Betti distributions of certain quasihomogeneous Hilbert schemes are of Gumbel type, extending previous results and providing new asymptotic formulas for partition counts with parts divisible by a fixed integer.

Contribution

It generalizes the Gumbel distribution result for Betti numbers to quasihomogeneous Hilbert schemes with torus actions and derives new asymptotic formulas for specific partition counts.

Findings

Limiting Betti distributions are Gumbel type for these Hilbert schemes.

Derived asymptotic formula for the number of partitions with parts divisible by A.

Extended Erd"H{o}s-Lehner results to partitions with fixed multiples.

Abstract

Hausel and Rodriguez-Villegas recently observed that work of G\"ottsche, combined with a classical result of Erd\H{o}s and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes $(\mathbb{C}^2)^{[n]}$ on $n$ points, as $n\rightarrow +\infty,$ is a \textit{Gumbel distribution}. In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes $((\mathbb{C}^2)^{[n]})^{T_{\alpha,\beta}}$ that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erd\H{o}s and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer $A\geq 2.$ Furthermore, if $p_k(A;n)$ denotes the number of partitions of $n$ with exactly $k$ parts that are multiples of $A$, then we obtain the asymptotic $$ p_k(A,n)\sim \frac{24^{\frac k2-\frac14}(n-Ak)^{\frac k2-\frac34}}{\sqrt2\left(1-\frac1A\right)^{\frac k2-\frac14}k!A^{k+\frac12}(2\pi)^k}e^{2\pi\sqrt{\frac1{6}\left(1-\frac1A\right)(n-Ak)}}, $$ a result which is of independent interest.