Nagata’s conjecture and maximal Seshadri constants

1 · The essentials

How much can a plane curve pass through a collection of points, counting repeated passage at each point? The main manuscript gives a strict limit when there are at least ten complex points chosen in the special mathematical sense of “very general.” The total count must be less than the curve’s degree multiplied by the square root of the number of points. The counts can differ from point to point, and the statement includes curves made of several pieces or containing repeated pieces. This main result has a Lean formalization. Companion manuscripts address a related question about how much geometric positivity can be shared among many points, and state that the largest allowed value is reached once there are sufficiently many points. The complex-surface companion also has a Lean formalization; the supplied source does not identify formalizations for the higher-dimensional or positive-characteristic companions.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, think of a curve’s degree as a budget and its repeated passage through each chosen point as spending from that budget. The main result says that, for the specified point choices, the total spending stays strictly below a limit determined by the budget and the number of points.

Why it matters

The main manuscript establishes the stated strict, nonhomogeneous form of Nagata’s conjecture, allowing different multiplicities at different points. The surface companion resolves the qualitative Nagata–Biran conjecture for surfaces. The other companions extend the maximal-value conclusion to higher dimensions over the complex numbers and to geometric generic tuples in positive characteristic.

2 · Going further

The plane-curve problem concerns a nonzero effective curve of degree $d$ with multiplicities $m_{i}$ at $r \geq 10$ very general complex points. The main manuscript establishes $\sum_{i} m_{i} < d r$ , simultaneously for all such curves. Its scope explicitly includes reducible and nonreduced curves, and the multiplicities need not be equal. The manuscript identifies this as Nagata’s conjecture in its strict, nonhomogeneous form and has a Lean formalization.

The complex-surface companion considers a smooth integral projective surface $S$ and an ample line bundle $L$ . It establishes that the multipoint Seshadri constant at $r$ very general points equals $L^{2} / r$ for every sufficiently large integer $r$ . This manuscript also has a Lean formalization and identifies its conclusion as the qualitative Nagata–Biran conjecture for surfaces.

For a smooth integral complex projective variety $X$ of dimension $n \geq 3$ , the third manuscript shows that there is a threshold $r_{0} (X, L)$ such that the ordinary multipoint Seshadri constant equals the volume bound $(L^{n} / r)^{1/ n}$ whenever $r \geq r_{0}$ . The fourth manuscript shows the same conclusion over an algebraically closed field of positive characteristic, at the geometric generic tuple rather than at very general complex points; it also includes dimension two. These are described as qualitative Nagata–Biran–Szemberg assertions in their respective settings.

The shared theme is a strict multiplicity bound for plane curves and attainment of the volume bound for multipoint Seshadri constants in broader settings. The supplied abstracts state the conclusions but do not describe the arguments, so they do not support an account of a proof strategy or of logical dependencies between the companion manuscripts.

Useful notions

Plane curves, degree, and multiplicity
Very general points and geometric generic tuples
Smooth projective varieties and dimension
Ample line bundles and multipoint Seshadri constants

What was known before / what is new

Before

The source frames the plane-curve statement as Nagata’s conjecture and the surface statement as the qualitative Nagata–Biran conjecture. It names the higher-dimensional assertions after Nagata–Biran–Szemberg and identifies the target value as the volume bound. It gives no account of earlier partial results or methods.

New

The main result supplies the strict inequality for every nonzero effective plane curve at at least ten very general complex points, with arbitrary multiplicities. The surface companion establishes maximal multipoint Seshadri constants for every sufficiently large number of very general points. The remaining manuscripts show the corresponding maximal-value conclusions in higher complex dimensions and in positive characteristic under the stated point-choice conditions.

Limits and assumptions

The plane statement requires very general complex points and r ≥ 10; it is not a statement about arbitrary point configurations. The Seshadri statements require smooth integral projective varieties and ample line bundles, and hold only for sufficiently large r, with a threshold depending on (X,L). In positive characteristic, the stated conclusion concerns geometric generic tuples. No explicit thresholds or argument details are supplied. The main and complex-surface manuscripts have Lean formalizations; the source does not specify their detailed formalization scope or identify formalizations for the other two manuscripts. It does not list further open questions.

Mini glossary

Total multiplicity: The sum of the multiplicities m_i assigned to the curve at the selected points.
Nonhomogeneous form: The form of the plane-curve assertion that allows the multiplicities at different points to differ.
Volume bound: The value $(L^{n} / r)^{1/ n}$ attained by the multipoint Seshadri constants in the stated companion results.
Threshold r_0(X,L): A number depending on the variety and line bundle beyond which the stated maximal-value conclusion holds for every integer r.

Keywords : Nagata’s conjecture · Multipoint Seshadri constants · Volume bound

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves Nagata's strict inequality $\sum_{i} m_{i} < d r$ 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.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/039.md

Manuscripts in this family

★

Nagata's conjecture for plane curves

23/09/2026

Verified by Lean

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.

PDF GitHub folder Lean formalization

Maximal Seshadri constants on arbitrary polarized surfaces

23/09/2026

Verified by Lean

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

PDF GitHub folder Lean formalization

Maximal Multipoint Seshadri Constants in Higher Dimensions

05/10/2026

To be verified

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 $r \geq r_{0}$ , 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.

PDF GitHub folder

Maximal multipoint Seshadri constants in positive characteristic

05/10/2026

To be verified

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.

PDF GitHub folder