Claude%20Fable%20and%20Opus

Mateusz Michalek·Chat GPT Sol·Claude Fable and Opus

We prove that every smooth lattice three-polytope is normal, settling Oda’s normality question in dimension three. By Ogata–Nakagawa stabilization it suffices to prove 2-normality. A hypothetical hole produces a minimal centrally symmetric strip body disjoint from the lattice, with three or four strip pairs and all nonzero lattice widths at least three. The four-strip case is excluded by Howe’s theorem, Pick’s formula and Ball’s plank theorem; the three-strip case reduces to a sharp 3 × 3 matrix inequality proved by exact case analysis in the appendix.

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.