Nagata conjecture

A line can pass through many points if they are all lined up. For points without such special arrangements, a curve must usually use a more complicated polynomial to pass through them, especially if it must vanish repeatedly at each point. How much degree is unavoidable?

Nagata’s conjecture gives a lower bound for curves through r≥10 very general points of the complex projective plane. Add the multiplicities, which count the required orders of vanishing at the points, then divide by √r: the curve’s degree must be at least that value. Very general excludes a countable collection of exceptional algebraic arrangements, not merely coinciding points. For sixteen multiplicities all equal to one, this bound gives degree at least four; it does not promise a degree-four curve exists.[1] [2]

Passing repeatedly through chosen points imposes vanishing conditions on a plane polynomial. Can those local demands force the predicted global degree bound for every curve when the points have no exceptional algebraic arrangement? If sixteen points all lie on one line, that line has degree 1 and multiplicity 1 at each; the bound $d\ge16/\sqrt{16}=4$ fails. This illustrates why the very-general condition is essential, not a counterexample within the conjecture.

Blowing up each chosen point replaces it by an exceptional curve, recording directions through that point. On $X=\operatorname{Bl}_{p_1,\ldots,p_r}\mathbb P^2_{\mathbb C}$, write $H$ for the pulled-back line class and $E_i$ for these exceptional classes. A plane curve with degree $d$ and multiplicities $m_i$ has strict-transform class $dH-\sum m_iE_i$. The proposed symmetric class is $L=\sqrt rH-\sum E_i$, for $r\ge10$. Its intersection with that curve is

$$L\cdot C=d\sqrt r-\sum_i m_i.$$

Nef means nonnegative intersection with every curve. Must $L$ be nef for very general points? This page keeps the non-strict nef formulation; a lower bound alone does not assert a curve exists at equality.[1] [2]

Square values of r are known; the bound remains open for general nonsquare values.[1]

The September 2026 irrational-constant construction uses a different divisor and does not prove this symmetric nefness target.[2] [3]

This would tell us when a proposed interpolation problem is impossible before searching for its polynomial. Repeated vanishing at scattered points can force a degree much higher than the same requests at specially aligned points. Understanding that limit connects local conditions at the chosen points to the global complexity of a curve.[1] [2]

A nef-cone boundary separates classes that are nonnegative on every curve from classes on which some curve detects negative intersection. Finding this boundary would therefore constrain both possible interpolation polynomials and positivity on the blown-up surface. Special configurations can make many vanishing conditions dependent, as the collinear example demonstrates; the conjecture says the very-general surface avoids the resulting low-degree curves below the symmetric threshold. This links local multiplicity requests to the global geometry of all curve classes, without turning a numerical lower bound into an existence theorem.[1] [2]