Which monomial subschemes are limits of smooth subschemes? 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:

When is $X$ in the closure of the locus in the Hilbert scheme parameterizing smooth subvarieties?

I'm particularly curious about the case when $X$ is zero dimensional, about which I would also hope more is known than in general...
 A: 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. 
A: Will completely answered the question in dimension $0$.  In dimension > 0, not all monomial ideals are flat limits of ideals of smooth schemes. For instance, consider the union $X$ of two coordinate $2$-planes in $\mathbb{P}^4$ that intersect in a single point.  By considering a generic hyperplane section (which is disconnected) and using Zariski's Main Theorem / Stein Factorization / principle of connectedness, this cannot be a specialization of a smooth, irreducible scheme.  However, the Hilbert polynomial of a union of two skew $2$-planes is strictly larger than the Hilbert polynomial of $X$.  
$\textbf{Edit}.$  I guess the argument is even simpler than I made it out to be: there is no such thing as a union of two skew $2$-planes in $\mathbb{P}^4$.  The same argument would work in $\mathbb{P}^5$, where now the Hilbert polynomial comparison is relevant.
