Episode 80 · The Machine's Theorems
The Classification Machines: Birational Geometry's Big Programme
Part 12 of "The Machine's Theorems", a lighter episode. Farah and Augustin use a refinery's pig-cleaning campaign and a sculptor's chisel to explain birational geometry: canonical divisors, contractions, flips, termination and abundance. We then read OpenAI's manuscripts on termination for fourfolds (family 056, including six Kähler manuscripts revised on 7 October), log abundance in every dimension (034) and in positive characteristic (035), Fujita's freeness conjecture (038), Nagata's conjecture (039, Lean-listed), P = W for fixed determinant (043) and a counterexample to Griffiths' positivity conjecture (050, Lean-listed). Every claim is labelled by tier.
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Inside this episode
- Cold open: the sculptor and the pig
- Same shape, different clothes
- A worked example: the plane with a point blown up
- Kodaira dimension: the three-way split
- The canonical divisor: a curvature compass
- Flips: cutting and re-gluing
- What a flip looks like
- The road to termination
- What family fifty-six claims
- Abundance: does the clean shape fibre nicely?
- What families thirty-four and thirty-five claim
- Why positive characteristic is harder
- Halfway: the tour so far
- Fujita: how many twists until the bundle is free?
- How you would check a short proof of a famous conjecture
- Nagata: how special can a curve be?
- Seshadri constants: positivity measured at a point
- P equals W: two filtrations, one truth
- Griffiths: a counterexample with an explicit bundle
- The machine in the cathedral
- Why it matters to an engineer
- Closing: the pig comes out
Production credits & AI disclosure
Scripts generated with Claude Opus. Synthetic narration generated with Edge TTS. Reviewed, listened to, and approved by Sankalp Thakur.