Twenty-one points in three-space
Mateusz Michalek·Chat GPT Astra and Sol·Claude Fable and Opus
Abstract
We prove that every finite subscheme of length twenty-one in complex affine three-space is a limit of distinct points. So the Hilbert scheme of twenty-one points in is irreducible, and the least number of points for which it is reducible lies between twenty-two and seventy-eight. The proof continues the induction of Nineteen points and Twenty points in three-space, whose main tool is linkage: a scheme of length d in a complete intersection of length less than 2d is linked to a shorter one. The new tool is linkage at infinity: we deform the tails of a scheme while magnifying it, and link the limit, which keeps part of the tails, instead of the tangent cone. When the tangent cone lies on a quadric, we move one equation in a flat family to cap the cost of its special directions. Formalisation in lean will be provided soon.Discussion
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