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@JasonStarr: Can I say that the form $\beta_r$ is restricted from the one on $\mathrm{Hilb}^{rg}(\mathbb{A}^2)$ onto this $G$-invariant smooth subvariety?
@JasonStarr: Thanks! This is what I had in mind. But I am not comfortable to say that $\beta = dx \wedge dy$ is $G$-invariant, since it is only so up to a scalar multiple.
@JasonStarr: I guess another way of phrasing the statement is that the cardinality of a minimal generating set of any non-free maximal Cohen-Macaulay module over the invariant ring is twice the rank of the module.
