Nagata’s conjecture and the cost of imposing many multiple points

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Family 039 · Algebraic and complex geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-12

1. The question

How complicated must a plane curve be to pass through many prescribed points with high multiplicity? A line can pass through two general points, while a cubic has more freedom. Requiring not only passage but repeated vanishing consumes additional freedom, and the interaction between these conditions can be subtle.

Nagata’s conjecture predicts the exact square-root scale once there are at least ten very general points. Family 039 claims its strict form, allowing different multiplicities at different points. Three companions broaden the question to local positivity on arbitrary polarized varieties and to fields of positive characteristic.

The broader claims are eventual exact equalities: for every point count beyond a threshold depending on the variety and polarization, a Seshadri constant reaches its volume upper bound. That is stronger than approaching the bound as the number of points grows, but it does not give the same sharp threshold ten in all settings.

2. Background and essential definitions

Multiple points as linear conditions

A degree-$d$ plane curve is the zero set of a homogeneous polynomial in three variables. Its coefficient space has dimension $\binom{d+2}{2}$. At a point, multiplicity at least $m$ means that all local terms of total degree below $m$ vanish. On a smooth surface this imposes at most $\binom{m+1}{2}$ linear conditions.

For $r$ points with equal multiplicity $m$, the leading terms of these counts balance when

$$\frac{d^2}{2}\approx\frac{rm^2}{2}, \qquad\text{or}\qquad d\approx m\sqrt r.$$

The count explains the predicted scale, but does not prove independence of the conditions. Special configurations, unexpected components, or dependencies can allow a curve even when a naive dimension count predicts none.

An effective curve may be reducible, and components may repeat. Multiplicity includes these repetitions. A nonzero effective curve has positive degree; no irreducibility assumption is made in the plane theorem.

“Very general” is a simultaneous condition

A property holds at very general tuples if its exceptional tuples lie in a countable union of proper algebraic closed subsets of the configuration space. Over $\mathbb C$, such a complement is nonempty. This is stronger than avoiding one obvious special configuration, such as collinearity.

Here the exceptional set must work simultaneously for every degree and every multiplicity vector. A new exceptional set chosen separately for each curve after seeing the tuple would not give the desired statement.

In positive characteristic, the algebraic closure of a finite field is countable. The companion therefore states its result at a geometric generic tuple: the universal tuple over an algebraic closure of the configuration space’s function field. This gives a precise notion of unrestricted algebraic general position without claiming the existence of a suitable tuple over a countable constant field itself.

Seshadri constants measure local positivity

Let $X$ be a smooth projective $n$-fold and $L$ an ample line bundle. Ampleness means some positive power defines a projective embedding. For distinct points $p_1,\ldots,p_r$, the ordinary multipoint Seshadri constant is

$$\varepsilon(X,L;p_1,\ldots,p_r) =\inf_C\frac{L\cdot C}{\sum_{i=1}^r\operatorname{mult}_{p_i}C},$$

where $C$ runs over integral curves meeting at least one marked point. The numerator is the degree of $L$ on the curve; the denominator counts how strongly the curve concentrates at the points. Multiplicity is zero at a point outside the curve.

Blowing up the points replaces each point by its tangent directions, giving exceptional divisors $E_i$ and a map $\pi:Y\to X$. The same constant is the largest $t\geq0$ for which

$$\pi^*L-t\sum_{i=1}^rE_i$$

is nef, meaning nonnegative on every curve. Thus it measures how much equal local weight can be subtracted while retaining numerical positivity.

If $V=L^n$ is the top self-intersection, the volume bound is

$$\varepsilon(X,L;p_1,\ldots,p_r)\leq(V/r)^{1/n}.$$

At that value, the displayed blowup class has top self-intersection zero. In higher dimensions these manuscripts still use the ordinary constant defined by curves, not a different invariant defined by higher-dimensional cycles.

3. Development history and previous partial results

Nagata’s 1959 work on Hilbert’s fourteenth problem led to the plane interpolation conjecture and proved its square-point-count cases from sixteen onward. Square counts admit configurations and estimates that do not automatically extend to nonsquares.

Later degeneration methods and rational lower bounds brought the constants close to the predicted value. For ten very general plane points, the bound $d/m\geq117/37$ is a notable predecessor discussed in M1. Since $117/37$ is close to $\sqrt{10}$, the remaining numerical gap is small, but closing it must work for arbitrarily large multiplicities. Checking finitely many multiplicities does not suffice.

The problem extends naturally to arbitrary polarized surfaces through Seshadri constants. Biran’s symplectic packing theorem provides a related stability phenomenon for many equal balls. However, symplectic packing allows geometric flexibility that does not itself prove nefness on blowups of a fixed complex variety. The algebraic and symplectic conclusions must be distinguished.

Harbourne established important square-volume and asymptotic cases. Roé–Ross related constants for different point counts through product inequalities. In higher dimensions, Küchle obtained asymptotically optimal lower bounds. These results explain why the volume bound is the right scale, but do not by themselves imply exact equality for every sufficiently large integer count.

The interpolation techniques used here have direct precedents in Ciliberto–Miranda’s degenerations, Evain’s infinitesimal limits, and Dumnicki’s monomial diagrams. Valuation descriptions of linear series and finite-degree jet comparisons supply a further route from geometry to exponent counting. M1–M4 identify these antecedents while distinguishing their new uniformity claims.

4. What the new manuscripts claim

M1: the strict plane inequality

For every integer $r\geq10$, M1 claims that at very general complex plane points, every nonzero effective curve of degree $d$, with multiplicities at least nonnegative integers $m_i$, satisfies

$$\sum_{i=1}^r m_i<d\sqrt r.$$

The inequality is strict, permits unequal multiplicities, and includes reducible and nonreduced curves. At nine points a cubic always exists, so the strict statement cannot start there: its total multiplicity is at least nine, equal to $3\sqrt9$.

For the plane polarization $\mathcal O_{\mathbb P^2}(1)$, the consequence is

$$\varepsilon(\mathbb P^2,\mathcal O(1);p_1,\ldots,p_r)=\frac1{\sqrt r}.$$

When $r$ is nonsquare this is irrational. Strictness for each individual curve is consistent with equality of the infimum: curve ratios can approach a value without attaining it. The associated real blowup class has square zero and positive intersection with every curve; that does not make it ample.

M2: arbitrary polarized surfaces

For every smooth integral complex projective surface $S$ and ample $L$, M2 claims a threshold $r_0(S,L)$ such that, for each integer $r\geq r_0$,

$$\varepsilon(S,L;p_1,\ldots,p_r)=\sqrt{L^2/r}$$

at very general tuples. The threshold may depend on the polarized surface, and no sharp explicit universal value is asserted. The plane theorem’s strict inequality and threshold ten are a separate refinement.

M3: higher complex dimensions

M3 claims that for every smooth integral complex projective variety of dimension $n\geq3$ and every ample $L$, all sufficiently large point counts satisfy

$$\varepsilon(X,L;p_1,\ldots,p_r)=(L^n/r)^{1/n}$$

at very general tuples. Again, the threshold depends on $(X,L)$. This is exact for each such $r$, rather than a limiting ratio as $r\to\infty$. The paper adapts the surface method but states that its proof does not use the surface maximality theorem as an input.

M4: positive characteristic

M4 asserts the same eventual maximality at geometric generic tuples for smooth integral projective varieties over any algebraically closed field of characteristic $p>0$, in dimensions at least two. Characteristic two is included.

The claim is qualitative in the point count and uses the ordinary curve-defined constant. It does not assert the plane threshold ten or strict Nagata inequality for every positive characteristic. The chosen generic-field formulation is part of the statement, not merely a change in terminology.

5. Why the advance matters

Importance: 5/5, provisional. The plane claim resolves a central interpolation conjecture with exact constants, while the companions assert broad maximal local positivity for arbitrary polarizations and dimensions. The family connects curve geometry, linear systems, blowup cones, and packing-type questions.

Exact irrational Seshadri constants are particularly informative: every individual curve contributes a rational degree-to-multiplicity ratio, yet their infimum can be irrational. Such a boundary is not detected by one exceptional curve computing the minimum.

The result also distinguishes two kinds of uniformity. The plane bound works at every count from ten onward. The general statements allow a variety-dependent threshold but must work above it for every degree and every jet order.

6. The main idea

Turn a geometric counterexample into a rank problem

M1 uses closed incidence conditions to turn a hypothetical failure at very general points into systems existing at all configurations. Multiplying over permutations equalizes the multiplicities. Points are then moved toward a smooth auxiliary curve, producing a polynomial on its normal bundle.

For nonsquare point counts, a cubic supplies an elliptic curve. After further collisions, the required vanishing implies a kernel in a finite jet-evaluation matrix. A theta-function degeneration gives explicit limiting bases whose derivatives become powers of integer exponents. The question becomes whether a concrete matrix has full column rank.

Counting only the total number of exponents is insufficient. M1 groups them into rows and uses interpolation row by row. Polygon widths control the possible row sizes, and the strict gap below $\sqrt r$ absorbs rounding errors. Full rank contradicts the required kernel.

Surface positivity through injective jet tests

A jet records Taylor coefficients up to a specified order. An injective jet test means that a section vanishing to the required order at every test point must be zero. M2 constructs such tests for sections whose equal vanishing would exceed the critical Seshadri slope.

A nodal curve on the surface constrains leading exponents to polygons. Changes of lattice direction and finite torus subgroups adapt the test to every sufficiently large integer $r$, not only square grids. The subgroup and shape are chosen before the arbitrarily large jet orders.

Open rank conditions transfer successful tests to very general tuples. A further intersection and section argument turns exclusion of equal-order sections into a bound against all curves, including those with unequal point multiplicities.

Higher dimensions through surjective tests

M3 instead constructs surjective jet maps: enough sections to prescribe all required Taylor data independently at the marked points. Jet separation through order $a$ by sections of $kL$ gives a lower bound $\varepsilon\geq a/k$.

Leading exponents form finite sets $S_k$ with the multiplication property

$$S_k+S_\ell\subseteq S_{k+\ell}.$$

The manuscript reshapes the containing simplex two coordinates at a time, preserving both the relevant section counts and this multiplication rule. Full asymptotic density can then be upgraded to exact filling of compact regions inside the simplex. One-variable Hermite interpolation along its long direction separates jets at $r$ points.

M4 replaces analytic limits by formal algebraic substitutions and uses interpolation through the Chinese remainder theorem, valid in every characteristic. Its nodal equation retains two distinct branches even in characteristic two. Each finite test holds on a nonempty open set, and therefore at the geometric generic tuple.

7. Scope, limitations, and open questions

Very general position is essential. Special configurations can lie on low-degree curves and have smaller constants. The results do not furnish the same bounds for an arbitrary tuple supplied by coordinates.

The surface and higher-dimensional statements give eventual exact equality with a threshold depending on the polarized variety. They do not determine that threshold, nor do they give exact values for every small point count. M1’s strict plane conclusion is stronger in this particular respect.

The positive-characteristic result is stated at geometric generic tuples. Over a countable algebraically closed field, it should not be paraphrased as a guarantee about constant-field points outside a countable exceptional union.

Further questions include effective thresholds, special point configurations, and weighted analogues with unequal subtraction parameters. The family permits unequal multiplicities of test curves while defining an equal-weight Seshadri constant; these are different notions of “unequal.” The manuscripts are reported as stated without proof verification.

8. Sources and further reading

M1 §1 states strict Nagata and §§2–5 develop the collision and rank argument. M2 §§1 and 4–5 explain the surface threshold, torus tests, and passage to nefness. M3 §§1 and 4–6 give reshaping, interior filling, and higher-dimensional jet separation. M4 §§1–2 specify the geometric generic convention and characteristic-independent transfers.

The linked Nagata, Biran, Roé–Ross, and Evain sources give primary historical context. An optional introduction to Seshadri constants expands on the local-positivity interpretation.

Source manuscripts

• M1. OpenAI, Nagata’s conjecture for plane curves (September 23, 2026).
• M2. OpenAI, Maximal Seshadri constants on arbitrary polarized surfaces (September 23, 2026).
• M3. OpenAI, Maximal Multipoint Seshadri Constants in Higher Dimensions (October 5, 2026).
• M4. OpenAI, Maximal multipoint Seshadri constants in positive characteristic (October 5, 2026).

Sources consulted on 12 October 2026 at repository revision fd4aeeb. Download the manuscript citations.