OpenAI 给出 Nagata 猜想的简洁证明，UMass Amherst 讲义详解
On OpenAI's proof of the Nagata conjecture
OpenAI 证明了几十年悬而未决的 Nagata 猜想，这篇讲义把证明拆到非专家也能跟上，值得花半小时读读
arXiv 讲义整理了 OpenAI 对 Nagata 猜想的证明思路。证明通过标准归约到 normal cone，将 fat points 塌缩为单一 monomial scheme，问题化为光滑平面三次曲线的 normal ruled surface 上的插值。核心新步骤是利用三次曲线退化出 node 时 theta 函数的渐近行为，配合插值对偶，把问题进一步化归为三角格点上的插值。同一计算还确定了权重比大于 9 时加权 blow-up 的 nef 与 pseudoeffective cone。
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.
