Hilbert point and Hilbert stability

Question

For $X\in \mathbb{P}^N$ a closed subscheme, one can consider the m-th Hilbert point $$ [X]_m=[\bigwedge^{h^0(X, \mathcal{O}(m))}H^0(\mathbb{P}^N, \mathcal{O}(m))\to \bigwedge^{h^0(X, \mathcal{O}(m))}H^0(X, \mathcal{O}(m))]\in \mathbb{P}(W_m). $$

The scheme $X$ is said to be Hilbert stable (resp. Hilbert semistable) if m-th Hilbert points of $X$ are all GIT stable (resp. GIT semistable) for $m>>1$.

My question is: Can you give an example that $[X]_m$ is GIT stable for some $m$ but $X$ is not Hilbert stable?

Answer 1

This phenomenon happens already for curves. Let $C$ be a curve in $\mathbb{P}^2$ of degree $4$, which has an ordinary cusp, locally $y^2=x^3+\text{higher terms}$. Then $C$ is GIT stable, by 1.12 in Mumford's "Stability of projective varieties". On the other hand it is known that asymptotically Hilbert stable curves cannot have cusp singularities (they can have at most double points); this is also contained in Mumford's "Stability of projective varieties" Corollary 3.2.

I think it would be interesting to have a smooth example of your question (I cannot think of any offhand). It might be worth mentioning that there are plenty of examples of smooth varieties that are not asymptotically Hilbert stable, for example $Bl_p\mathbb{P^2}$ with respect to the anti-canonical polarisation. This follows as this variety is K-unstable, which in turn implies asymptotic Hilbert instability. To learn about this story see e.g. Ross-Thomas's "A study of the Hilbert-Mumford criterion for the stability of projective varieties". However I do not know of any such examples that are GIT stable under some embedding, as proving GIT stability is quite difficult.

Edit: if one is interested in Chow stability rather than Hilbert stability (i.e. one does GIT on the Chow variety rather than the Hilbert scheme), then there are smooth examples. Indeed, Keller-Ross give examples of Chow stable but asymptotically Chow unstable projective bundles: http://arxiv.org/pdf/1110.4489v3.pdf It seems to be an open problem to give a smooth example for Hilbert stability.