Say $X \subset \mathbb{P}^n$ is monomial, by which I mean that it's cut out by a homogenous ideal generated by monomials, or in other words it's a fixed point on the Hilbert scheme for the action of the big torus. What is known about the following question:



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> When is $X$ in the closure of the locus in the Hilbert scheme parameterizing smooth subvarieties?
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I'm particularly curious about the case when $X$ is zero dimensional, about which I would also hope more is known than in general...



Which monomial subschemes are limits of smooth subschemes?
In the zero-dimensional case, the scheme will certainly be supported only on the $n+1$ fixed points in $\mathbb P^n$ of the torus action, so we can work locally in a neighborhood of each point. Thus we can consider an affine monomial ideal.


Replace the monomial $ \prod\_i x\_i^{d\_i}$ with $\prod\_i (x\_i)(x\_i-t)(x\_i-2t)\dots (x\_i-d\_it)$, and form the new ideal generated by these. Because the scheme is zero-dimensional, you always have some monomial that is a power purely of $x\_i$ for each $i$, thus the scheme is a disjoint union of points, so smooth. As $t\to 0$ it converges to the monomial ideal in a finite flat way. 


