On OpenAI's proof of the Nagata conjecture

These notes are an expanded account of my talk at UMass Amherst on OpenAI's remarkably simple proof of the Nagata conjecture. I hope they will be useful, especially to non-experts. The standard reduction to the normal cone and the collision of fat points into one monomial scheme reduce the problem to interpolation of monomial schemes on the normal ruled surface of a smooth plane cubic. The novelty is that the study of the asymptotics of theta functions as the cubic develops a node, together with interpolation duality, reduces the problem further to interpolation on a triangle of lattice points, which is straightforward. The same calculation determines the nef and pseudoeffective cones of a weighted blow-up of ${\Bbb P}^2$ at one point for very general local parameters when the ratio of weights is greater than nine.