Anthropic%20Fable%20and%20Opus

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.