Nagata’s conjecture and maximal Seshadri constants

Proves Nagata's strict inequality \sum_i m_i\lt d\sqrt r for every nonzero effective plane curve of degree d through r ≥ 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 ≥ 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)

Manuscripts (4)

Nagata's conjecture for plane curves

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:Nagatas-Conjecture-for-Plane-Curves-September-23-2026}

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\sqrt r. The inequality holds simultaneously for all curves, including reducible and nonreduced curves, and establishes Nagata's conjecture in its strict, nonhomogeneous form.

Maximal Seshadri constants on arbitrary polarized surfaces

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026}

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 \sqrt{L^2/r} for every sufficiently large integer r. This resolves positively the qualitative Nagata–Biran conjecture for surfaces.

Maximal Multipoint Seshadri Constants in Higher Dimensions

October 5, 2026 · source · cite as \cite{OAI:Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions-October-5-2026}

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 r_0=r_0(X,L) such that for every integer r\ge r_0, the ordinary multipoint Seshadri constant at r very general points equals the volume bound (L^n/r)^{1/n}. This establishes the qualitative Nagata–Biran–Szemberg assertion in these dimensions.

Maximal multipoint Seshadri constants in positive characteristic

October 5, 2026 · source · cite as \cite{OAI:Maximal-multipoint-Seshadri-constants-in-positive-characteristic-October-5-2026}

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\ge3. We prove that there is a threshold r_0=r_0(X,L) such that for every integer r\ge 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.

Result 039 · Algebraic and complex geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.