Nagata’s conjecture and maximal Seshadri constants

What OpenAI claims

Proves Nagata's strict inequality ∑_i m_i < d√r for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities mi. It also proves maximal multipoint Seshadri constants (Lⁿ/r)^1/n for every smooth polarized projective variety of dimension n ≥ 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. (Lean)

Papers (4)

1. Nagata's conjecture for plane curves

   23 September 2026 · Preprint, not peer reviewed · PDF

   We prove that a nonzero effective plane curve of degree d at r ≥ 10 very general complex points has total multiplicity strictly less than d√r. The inequality holds simultaneously for all curves, including reducible and nonreduced curves, and establishes Nagata's conjecture in its strict, nonhomogeneous form.

2. Maximal Seshadri constants on arbitrary polarized surfaces

   23 September 2026 · Preprint, not peer reviewed · PDF

   We prove that, for every smooth integral complex projective surface S and every ample line bundle L, the multipoint Seshadri constant at r very general points equals √(L²/r) for every sufficiently large integer r. This resolves positively the qualitative Nagata–Biran conjecture for surfaces.

3. Maximal Multipoint Seshadri Constants in Higher Dimensions

   5 October 2026 · Preprint, not peer reviewed · PDF

   Let L be an ample line bundle on a smooth integral complex projective variety X of dimension n ≥ 3. We prove that there is a threshold r₀=r₀(X,L) such that for every integer r ≥ r₀, the ordinary multipoint Seshadri constant at r very general points equals the volume bound (Lⁿ/r)^1/n. This establishes the qualitative Nagata–Biran–Szemberg assertion in these dimensions.

4. Maximal multipoint Seshadri constants in positive characteristic

   5 October 2026 · Preprint, not peer reviewed · PDF

   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 ≥ 3. We prove that there is a threshold r₀=r₀(X,L) such that for every integer r ≥ r₀, the ordinary multipoint Seshadri constant at the geometric generic tuple of r points equals the volume bound (Lⁿ/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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.