I'm trying to understand the punctual Hilbert scheme of a smooth complex algebraic surface, and I think I'll have a much clearer picture of it if I understand the case of $n$ points in the affine plane very concretely.

Here's the idea. Let $X$ be the complex affine plane, or $\mathbb{C}^2$ for us dummies. A point in the **punctual Hilbert scheme** $M_{X, n}$ is an ideal $J$ in the polynomial ring $\mathbb{C}[x,y]$ for which the dimension of $\mathbb{C}[x,y]/J$ is $n$. There's a way to make the punctual Hilbert scheme into a smooth complex variety.

The easiest way to get a point in $M_{X,n}$ is to take $n$ distinct points $p_1, \dots, p_n$ in $\mathbb{C}^2$ and let $J$ consist of polynomial functions on $\mathbb{C}^2$ that vanish at all these points. This simple recipe works generically: we get an open dense set in the punctual Hilbert scheme this way. All the fun happens in the non-generic situation where some of the points $p_1, \dots, p_n$ collide.

When $n = 2$, we get more points in the punctual Hilbert scheme this way: take just *one* point $p$, and let $J$ consist of polynomials that vanish at $p$ *and* whose derivative at $p$ vanishes in one chosen direction. It's easy to check that $J$ is an ideal, and the dimension of $\mathbb{C}[x,y]/J$ is 2 because we're imposing two equations. We can think of these points in the punctual Hilbert scheme as arising from 'pairs of infinitesimally nearby points' in the plane.

**Question 1.** Does this trick give all the remaining points in $M_{X,2}$?

I feel the answer must be yes. If so we can move on to this:

**Question 2.** What tricks do we need to get all the points in $M_{X,3}$?

Here are some tricks I'm guessing that we need:

We can take $J$ to consist of all polynomials that vanish at 3 distinct points.

We can take $J$ to consist of all polynomials that vanish at one point and vanish along with a chosen directional derivative at a second, distinct, point.

More interestingly, we can take $J$ to consist of all polynomials that vanish along with

*both*their first partial derivatives at a single point. Presumably these points in $M_{X,3}$ describe ways for 3 points in the plane to collide. I imagine that in the usual topology of $M_{X,3}$ as a complex manifold they are limits of situations where we have 3 points arranged in a triangle and the triangle shrinks to zero size while remaining the same shape, with one corner not moving. Apparently the shape of the triangle doesn't matter as long as the triangle is nondegenerate, i.e. the 3 points don't lie on a line.More interestingly still, we can take $J$ to consist of all polynomials that vanish at some point along with their first

*and second*directional derivatives in a chosen direction at a single point. Apparently these points in $M_{X,3}$ describe subtler ways for 3 points in the plane to collide: namely, limits of situations where we have 3 points in a line, all moving closer and closer to a chosen point on that line.

Perhaps these tricks give all of $M_{X,3}$, but perhaps I'm leaving something out (or making a still worse mistake). Assuming I'm on the right track, the obvious question is:

**Question 3.** What tricks of this sort give all the points in $M_{X,n}$ for arbitrary $n$?

We seem to be getting some sort of stratification of $M_{X,n}$. How can we keep track of these strata? There might be some combinatorial way to name them all. It reminds me a bit of the Fulton–MacPherson compactification of a configuration space, but it seems different.

Perhaps everything I need to know is lurking in here:

- José Bertin, The punctual Hilbert scheme: an introduction.

but it will take some work to dig it out.