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.

History
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 $k[x_{1}, ..., x_{n}]$ 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.

Statement

 * Nagata Conjecture. Suppose $p_{1}, ..., p_{r}$ are very general points in $P^{2}$ and that $m_{1}, ..., m_{r}$ are given positive integers. Then for $r > 9$ any curve $C$ in $P^{2}$ 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 condition $r > 9$ is necessary: The cases $r > 9$ and $r ≤ 9$ are distinguished by whether or not the anti-canonical bundle on the blowup of $P^{2}$ at a collection of $r$ points is nef. In the case where $r ≤ 9$, the cone theorem essentially gives a complete description of the cone of curves of the blow-up of the plane.

Current status
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.