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

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 $\mathbb{C}^3$ 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.

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

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 $\mathbb{C}^3$ 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.

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

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.

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

We show that every finite subscheme of length at most fourteen in complex affine three-space is a limit of reduced subschemes. Equivalently, the Hilbert schemes of twelve, thirteen and fourteen points in $\mathbb{C}^3$ are irreducible. The proof uses two ways of smoothing a finite scheme: placing it on a surface with a rational double point, and splitting off a single point. Which of the two applies is decided mainly by the quadrics that vanish on the tangent cone. For one Hilbert function, (1, 3, 6, 4), we use a computer-assisted theorem of Douvropoulos, Jelisiejew, Nødland and Teitler on very compressed algebras.