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 over some field 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 are very general points in and that are given positive integers. Then for any curve in that passes through each of the points with multiplicity must satisfy
\degC>
| 1 | |
| \sqrt{r |
The condition is necessary: The cases and are distinguished by whether or not the anti-canonical bundle on the blowup of at a collection of points is nef. In the case where, the cone theorem essentially gives a complete description of the cone of curves of the blow-up of the plane.
The only case when this is known to hold is when 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.