# 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 $k[x_1, \ldots x_n]$ 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.
More precisely suppose $p_1,\ldots,p_r$ are very general points in the projective plane $P^2$ and that $m_1,\ldots,m_r$ are given positive integers. The Nagata conjecture states that for $r > 9$ any curve $C$ in $P^2$ that passes through each of the points $p_i$ with multiplicity $m_i$ must satisfy
$\mathrm{deg}\, C > {\sum_{i=1}^r m_i \over \sqrt{r}}.$
The only case when this is known to hold is when $r$ is a perfect square (i.e. is of the form $r=s^2$ for some integer $s$), 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 $r \le 9$ are distinguished by whether or not the anti-canonical bundle on the blowup of $P^2$ at a collection of $r$ points is nef.