The qualitative Nagata-Biran-Szemberg conjecture: maximal multipoint Seshadri constants for many very general points

Let $X$ be a smooth projective variety of dimension $n$ with an ample line bundle $L$ , and let $\varepsilon(X,L;p)$ be the multipoint Seshadri constant at $r$ points, the infimum of $L\cdot C/\sum_i\mathrm{mult}_{p_i}C$ over curves $C$ . It is always at most $(L^n/r)^{1/n}$ . Generalizing Nagata's plane conjecture, and motivated by Biran's symplectic packing stability, Szemberg and coauthors conjectured that on surfaces equality holds at $r$ very general points for every sufficiently large $r$ (Strycharz-Szemberg and Szemberg 2004, in a stronger form with an explicit threshold; Syzdek-Szemberg 2010, Conjecture 4.3), and Roé and Ross stated the same eventual maximality in every dimension over uncountable algebraically closed fields (2009, Conjecture 1.3). Harbourne had it when $rL^2$ is a square. Is there, for every polarized $(X,L)$ , a threshold $r_0$ such that $\varepsilon(X,L;p)=(L^n/r)^{1/n}$ at very general $p$ for every $r\ge r_0$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Algebraic geometry; Seshadri constants and positivity
Posed by: Beata Strycharz-Szemberg and Tomasz Szemberg (surfaces, stronger form); Wioletta Syzdek and Tomasz Szemberg (qualitative form); Joaquim Roé and Julius Ross (all dimensions)
Year posed: 2004
Years open: 22y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 38 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Surfaces (Theorem 1.1, principal): for every smooth integral complex projective surface $S$ and ample $L$ there is $r_0(S,L)$ such that for every $r\ge r_0$ , at $r$ very general points, $L-\sqrt{L^2/r}\sum E_i$ is nef on the blowup, so $\varepsilon(S,L;p)=\sqrt{L^2/r}$ ; Corollary 5.5 extends this to uncountable algebraically closed fields of characteristic zero. Companion, $n\ge3$ over $\mathbb C$ : $\varepsilon(X,L;p)=(L^n/r)^{1/n}$ at very general points for every $r\ge r_0(X,L)$ . Companion, positive characteristic, $n\ge2$ : the same equality at the geometric generic tuple. Not shown: an explicit or uniform threshold, the higher-cycle Seshadri analogues Roé-Ross distinguish, or very general points (rather than the geometric generic tuple) in positive characteristic. The sharp plane case $r\ge10$ is the separate Nagata entry.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has three manuscripts on this question: surfaces (September 23, 2026), complex dimension at least three (October 5, 2026) and positive characteristic (October 5, 2026). The higher-dimensional paper says it adapts the planar mechanism and does not use the surface theorem as an input.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of each of the three manuscripts was read against the conjecture as stated by Syzdek-Szemberg and Roé-Ross. The proofs were not refereed. lean/formalization.yaml lists a main result for the surface manuscript only (comparator MaximalSeshadriConstants, declaration OAI.MaximalSeshadri.Geometry.maximalSeshadriConstants). Its statement, read here: for a smooth integral surface given as a closed subscheme of complex projective space and an ample line bundle $L$ , there is $r_0$ such that for every $r\ge r_0$ a countable family of proper closed sets of distinct $r$ -tuples, with a tuple outside all of them, has the property that outside them a point blowup exists on which the boundary class is nef and the Seshadri constant equals $\sqrt{L^2/r}$ . That is the headline claim for surfaces over $\mathbb C$ . Not rebuilt here. The dimension $n\ge3$ and positive-characteristic manuscripts are not formalized and are unreviewed; in positive characteristic the result is at the geometric generic tuple, not at very general points. The thresholds are not explicit, so the stronger 2004 form with a specified threshold is not addressed.

The qualitative Nagata-Biran-Szemberg conjecture: maximal multipoint Seshadri constants for many very general points

What was actually shown

What the AI did

Verification

Sources

Discussion