OpenAI · 039-04 · 2026-10-05 · Algebraic and complex geometry

Maximal multipoint Seshadri constants in positive characteristic

preprints/Maximal-multipoint-Seshadri-constants-in-positive-characteristic-October-5-2026

Family 039. Nagata's conjecture and maximal Seshadri constants.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves Nagata's strict inequality $\sum_{i} m_{i} < d r$ for every nonzero effective plane curve of degree $d$ through $r \geq 10$ very general complex points, with arbitrary multiplicities $m_{i}$ . It also proves maximal multipoint Seshadri constants $(L^{n} / r)^{1/ n}$ for every smooth polarized projective variety of dimension $n \geq 2$ and all sufficiently large $r$ : at very general points over $C$ , and at the geometric generic tuple over any algebraically closed field of positive characteristic.

Let L be an ample line bundle on a smooth integral projective variety X over an algebraically closed field of positive characteristic, with $dim X = n \geq 3$ . We prove that there is a threshold $r_{0} = r_{0} (X, L)$ such that for every integer $r \geq r_{0}$ , the ordinary multipoint Seshadri constant at the geometric generic tuple of r points equals the volume bound $(L^{n} / r)^{1/ n}$ . The same conclusion holds in dimension two. This establishes the positive-characteristic form of the qualitative Nagata–Biran–Szemberg assertion at geometric generic tuples.

Preprint source and Lean scope

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A Lean scope note is stored for family 039, but it does not name this preprint. It names 039-01, 039-02. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.