Rationality for generating series for Hilbert scheme of points of curves with toric singularities

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?

• (Do add links from one copy of the question to the other, please) – Mariano Suárez-Álvarez Feb 15 '13 at 20:37

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 $K_0(\mathrm{Var}_\mathbb{C})$.
• How does the fact that $X^[n]$, as defined in this question, restricts the support to the origin affect the answer? – Victor Protsak Jan 10 '18 at 16:33