Sections everywhere?

0:00Sections that never vanish

Take a smooth projective variety X of dimension n, and an ample line bundle L on it. A line bundle is globally generated if, at every point, some global section is nonzero there. Ample bundles can fail this badly: L itself may have no nonzero sections at all. But add enough copies of L to the canonical bundle K, and sections appear. In 1985, Takao Fujita conjectured how many copies suffice: K + mL should be globally generated for every m at least n + 1. On a curve this is Riemann–Roch. K + 2L has degree at least 2g, and every line bundle of degree at least 2g is globally generated. The bound can’t be lowered. On n-dimensional projective space, with L the hyperplane bundle, the canonical bundle is O(−n − 1). So K + (n + 1)L is trivial, generated by a constant section, while K + nL is O(−1), which has no nonzero sections at all.

1:05Forty years of bounds

Reider proved the conjecture for surfaces in 1988. Ein and Lazarsfeld did threefolds in 1993, Kawamata fourfolds in 1997, and Ye and Zhu fivefolds. In general dimension, the bounds stayed far above n + 1: Angehrn and Siu got about n²/2, Heier on the order of n^(4/3), Ghidelli and Lacini on the order of n log log n, and a September preprint by Jingjun Han about 1.78 times n. Most of these proofs share one strategy. Build a divisor from sections of a multiple of L whose singularities concentrate at the point x, then use Kawamata–Viehweg vanishing to lift a section that is nonzero at x. The trouble is that the singular center may be a whole subvariety through x, possibly singular itself, and cutting it down to the point costs positivity. That cost is where the extra copies of L went.

2:03The claim

Result 038 is a single 25-page OpenAI preprint, dated September 23. It claims Fujita’s freeness conjecture in full: for every smooth complex projective variety of dimension n and every ample line bundle L, K + (n + 1)L is globally generated. Applying this to X times a projective space gives every larger m as well. The claim is global generation only. Fujita’s companion conjecture, that n + 2 copies give an embedding, is not addressed.

2:39How the proof goes

The proof argues by contradiction. Suppose every section of K + (n + 1)L vanishes at a point x. Expand the sections of high powers of L in Taylor series at x and record their leading terms. Counting lattice points, the average degree of those leading terms, divided by the power, is at least n/(n + 1), and projective space meets that exactly. At a base point, the first step shows the inequality must be strict, since in the equality case the blowup of x and vanishing already give a section that is nonzero there. Next comes a new minimization. Over all bases of sections, and all weighted ways of measuring vanishing whose center contains x, minimize an exponential average that rewards high vanishing orders and penalizes log discrepancy. The strict inequality keeps the minimum below one, and a compactness argument shows it is attained. The heart of the paper is that every minimizer is centered at x itself. If the center were a curve or a larger subvariety, varying the minimizer along it bounds a weighted moment of the leading terms from above. But restricting sections to the subvariety gives many of them, by an estimate that doesn’t depend on its degree or its singularities, and the Brunn–Minkowski inequality turns that into a lower bound that contradicts the upper one. With a minimizer centered at x, a small rational perturbation produces a divisor whose coefficient equals the log discrepancy exactly along a divisor S lying over x, and is strictly smaller on its neighbors. Kawamata–Viehweg vanishing lifts a nonzero value from S, and it descends to a section of K + (n + 1)L that is nonzero at x. So there was no base point.

4:29Where it stands

This is a preprint, not yet peer reviewed, with no Lean proof, and it has not been revised. It uses no other paper from the release, only standard tools: resolution of singularities, Kawamata–Viehweg vanishing and the semicontinuity of singularity exponents. We found no public assessment. If it holds, a conjecture from 1985 is settled at its sharp bound: on any smooth complex projective variety, adding n + 1 copies of an ample bundle to the canonical bundle gives, at every point, a section that doesn’t vanish there.