In Nakajima's book, "Lectures on Hilbert Schemes of Points on Surfaces", he gives an explicit description of the corresponding ideal for two points colliding in $\mathbb{C}^2$. This basically corresponds to the point itself plus the direction of the collision. I am trying to figure out an explicit ideal for when 3 points collide. Specifically I would like the generators of the ideal.
You should be able to think of three-point collision as two points colliding, then collide that "point" with the remaining point. But it shouldn't matter which two points you collide first, so I'm not sure what information I need. Maybe if I have three points collide at the origin, say, then I need the three vectors that those three points enter at?
Hint. A triple point is a local Artin $k$-algebra of dimension $3$. There are only two up to isomorphism, which are $$k[t]/t^3,\textrm{ and }k[x,y]/(x^2,xy,y^2).$$ In particular, every tripe point is planar. Essentially, these rings reflect the only two ways in which you can get a triple point: the first, as a $2$-jet, namely by colliding three points on a smooth curve (say, one of the coordinate axes); the second, by letting collide to the origin two points on different axes.