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Are there cases where all stable varieties in a Hilbert scheme are regular?

I have been wondering if there exist Hilbert polynomials $h(n)\in\mathbb{Q}[n]$, with the following property:

Every subscheme $Z\subset\mathbb{P}^{h(1)-1}$, whose Hilbert polynomial is $h$, and which is Hilbert stable is also regular, by which I mean that

• $H^i(Z,\mathscr{O}(n))=0$, for all $i>0$, and all $n>0$, and

• $\mathbb{C}[x_1,\ldots,x_{h(1)}]\to\bigoplus_{n=0}^\infty H^0(Z,\mathscr{O}(n))$ is surjective.

(Recall that $Z$ is Hilbert stable, if it is GIT-stable with respect to the action of $SL(h(1))$, on the Hilbert scheme $Hilb$ of all subschemes of $\mathbb{P}^{h(1)-1}$ with Hilbert polynomial $h$, and all linearizations coming from "sufficiently large" standard Grassmannian embeddings of $Hilb$.)

Thanks for considering my question. (I hope it makes sense.)

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 Are you excluding constant polynomials from consideration? – S. Carnahan♦ Sep 29 2011 at 18:47