The Hilbert scheme of the plane (or more generally, the resolution of the type A Kleinian singularity) has a surprising description as a Coulombbranch in the sense of Braverman, Finkelberg and Nakajima; this gives a rather odd looking presentation of the projective coordinate ring of this variety.  It turns out that this presentation is very well adapted to describing the tilting generators constructed using Bezrukavnikov and Kaledin's perspective; of course, there are many other interesting aspects of the combinatorics of coherent sheaves on these varieties, which hopefully the... Read more about Coherent sheaves on Hilbert schemes through the Coulomb lens