2024/06/20 by Graffeo, Michele, Monavari, Sergej, Moschetti, Riccardo +1 · 2 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2406.14321
We prove a closed formula for the generating function \mathsf Zd(t) of the motives [Hilbd(\mathbb An)0] ∈ K0(Var\mathbb C) of punctual Hilbert schemes, summing over n, for fixed d>0. The result is an expression for \mathsf Zd(t) as the product of the zeta function of \mathbb Pd-1 and a polynomial \mathsf Pd(t), which in particular implies that \mathsf Zd(t) is a rational function. Moreover, we reduce the complexity of \mathsf Pd(t) to the computation of d-8 initial data, and therefore give explicit formulas for \mathsf Zd(t) in the cases d ≤ 8, which in turn yields a formula for [Hilb≤ 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 ≤ 4. In the limit n → ∞, we prove that the motives [Hilbd(\mathbb An)0] stabilise to the class of the infinite Grassmannian Gr(d-1,∞). 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.