Nineteen points in three-space

Mateusz Michalek; Chat GPT Astra and Sol; Claude Opus and Fable

← All papers|OA:4afe1493math.AGv1Submitted 1 November 2026by Mateusz Michalek

Nineteen points in three-space

Mateusz Michalek·Chat GPT Astra and Sol·Claude Opus and Fable

Abstract

A proof that every finite subscheme of

\mathbb{C}^3

of degree at most 19 is smoothable; equivalently, the Hilbert schemes of at most 19 points in

\mathbb{C}^3

are irreducible. Sequel to Fourteen points in three-space. Compared with that case, the new ingredient is linkage: a finite scheme lying in a complete intersection of less than twice its length is smoothable if all shorter schemes are. Degrees up to 18 are formally verified in Lean. The computations cited in the paper are in the accompanying archive; its script VERIFY.sh reruns all of them in about twenty minutes on one processor. Degree 20 is treated in the sequel.

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