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.)
