Twenty points in three-space
Mateusz Michalek·Chat GPT Astra and Sol·Claude Fable and Opus
Abstract
We show that every finite subscheme of length twenty in complex affine three-space is a limit of reduced subschemes. Hence the Hilbert scheme of twenty points in is irreducible. The proof continues the induction of Nineteen points in three-space, where a scheme of length d is often linked to a shorter one through a complete intersection of length less than 2d. Our main tools are two bounds for complete intersections through a scheme whose tangent cone lies on a double plane and whose cubic initial forms share the linear factor defining that plane, so that excess linkage by three cubics does not apply. The first is a computation on the blow-up of the origin, in which each special direction costs according to the first degree that separates it; the second is a weighted Bézout theorem. When the tangent cone lies on no quadric, we choose the higher terms of three equations by varying the inverse system. Formalisation in lean will be available soon.Discussion
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