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 though a collection of very general points with prescribed multiplicity. 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 P² and that $m_1,\ldots,m_r$ are given positive integers. The Nagata conjecture states that for r > 9 any curve C in P² 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 = s² 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 P² at a collection of r points is nef.