Eventual maximal multipoint Seshadri constants on complex surfaces

For a smooth integral complex projective surface S with ample L, is the multipoint Seshadri constant at very general distinct r points eventually equal to its maximal value √(L²/r) for every sufficiently large integer r?

Why it matters and what counts as progress

Why it matters: Eventual maximality would strengthen square-case and asymptotic estimates to exact nef-boundary behavior.

What counts as progress: A theorem for every large r for each polarized surface, with the very-general-point qualification, or a surface and polarization with persistent nonmaximality.

How the problem developed

1959 · Conjecture [1]
Masayoshi Nagata Formulated the plane-curve conjecture for very general points that motivates the later maximality question for multipoint Seshadri constants.
23 Sept 2026 · Reported claim [1]
OpenAI internal model The companion preprint claims that on every smooth complex projective surface with an ample line bundle, the multipoint Seshadri constant is maximal at very general points for all sufficiently large numbers of points.

AI contribution

Maximal Seshadri constants on arbitrary polarized surfaces

OpenAI

For every smooth integral complex projective surface S with ample L, the manuscript claims that all sufficiently large r have maximal multipoint Seshadri constant ε(S,L;p)=√(L²/r) at very general distinct r-tuples, with a nef boundary divisor on the blowup.

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (Lean 4 kernel and Comparator)

Human understandingFew