Nagata's conjecture on curves

In mathematics, the Nagata conjecture on curves, named after Masayoshi Nagata, governs the minimal degree required for a plane algebraic curve to pass through a collection of very general points with prescribed multiplicities. Nagata arrived at the conjecture via work on the 14th problem of Hilbert, which asks whether the invariant ring of a linear group action on the polynomial ring k[x₁, ..., x_n] over some field k is finitely generated. Nagata published the conjecture in a 1959 paper in the American Journal of Mathematics, in which he presented a counterexample to Hilbert's 14th problem:

Nagata Conjecture. Suppose p₁, ..., p_r are very general points in P² and that m₁, ..., m_r are given positive integers. Then for r > 9 any curve C in P² that passes through each of the points p_i with multiplicity m_i must satisfy

$\deg C > \frac{1}{\sqrt{r}}\sum_{i=1}^r m_i.$

The only case when this is known to hold is when r is a perfect square, which was proved by Nagata. Despite much interest the other cases remain open. A more modern formulation of this conjecture is often given in terms of Seshadri constants and has been generalised to other surfaces under the name of the Nagata–Biran conjecture.

The condition r > 9 is easily seen to be necessary. The cases r > 9 and r ≤ 9 are distinguished by whether or not the anti-canonical bundle on the blowup of P² at a collection of r points is nef.

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