More concretely, let $I$ be a monimial ideal in $\mathbb{C}[x_1,\ldots,x_n]$ such that the ring $A=\mathbb{C}[x_1,\ldots,x_n]/I$ is one dimensional. Let $X=\operatorname{Spec}A$, and let $X^{[n]}$ be the punctual Hilbert scheme of points located at the origin. Then is it true that the generating series for Euler characteristic $\sum_{n}\chi(X^{[n]})t^n$ is rational?
1 Answer
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The answer is yes in the case that $X$ is reduced. In fact the following more general fact is true (see this recent preprint):
Theorem. Let $X$ be a reduced curve. Then $\sum_n \chi_{mot}(X^{[n]})t^n$ is a rational function where $\chi_{mot}$ is the universal Euler characteristic valued in the Grothendieck ring of varieties $K_0(\mathrm{Var}_\mathbb{C})$.

$\begingroup$ How does the fact that $X^[n]$, as defined in this question, restricts the support to the origin affect the answer? $\endgroup$ Jan 10, 2018 at 16:33

$\begingroup$ @VictorProtsak Right I should have addressed that. The series is still rational. In fact the way we prove the above theorem is by first reducing it to the series for Hilbert schemes supported on a singularity. $\endgroup$ Jan 10, 2018 at 16:42