The motive of the Hilbert scheme of points in all dimensions

Abstract

We prove a closed formula for the generating function ${\mathsf{Z}}_d(t)$ of the motives $[{\mathrm{Hilb}}^d({\mathbb{A}}^n{)}_0]\in K_0({\mathrm{Var}}_{\mathbb{C}})$ of punctual Hilbert schemes, summing over $n$, for fixed $d>0$. The result is an expression for ${\mathsf{Z}}_d(t)$ as the product of the zeta function of ${\mathbb{P}}^{d-1}$ and a polynomial ${\mathsf{P}}_d(t)$, which in particular implies that ${\mathsf{Z}}_d(t)$ is a rational function. Moreover, we reduce the complexity of ${\mathsf{P}}_d(t)$ to the computation of $d-8$ initial data, and therefore give explicit formulas for ${\mathsf{Z}}_d(t)$ in the cases $d\le 8$, which in turn yields a formula for $[{\mathrm{Hilb}}^{\le 8}(X)]$ for any smooth variety $X$. We perform a similar analysis for the Quot scheme of points, obtaining explicit formulas for the full generating function (summing over all ranks and dimensions) for $d\le 4$. In the limit $n\to \infty$, we prove that the motives $[{\mathrm{Hilb}}^d({\mathbb{A}}^n{)}_0]$ stabilise to the class of the infinite Grassmannian $\mathrm{Gr}(d-1,\infty )$. Finally, exploiting our geometric methods, we conjecture (and partially confirm) a structural result on the 'error' measuring the discrepancy between the count of higher dimensional partitions and MacMahon's famous guess.

55 pages, comments welcome

References in corpus (2)

• On the motive of the nested Quot scheme of points on a curve
• Hyperquot schemes on curves: virtual class and motivic invariants

Cited by in corpus (1)

• Invariants of nested Hilbert and Quot schemes on surfaces