Nagata's conjecture on curves

Template:For 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 Template:Math over some field Template:Mvar 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 Template:Math are very general points in Template:Math and that Template:Math are given positive integers. Then for Template:Math any curve Template:Mvar in Template:Math that passes through each of the points Template:Math with multiplicity Template:Math must satisfy

$\mathrm{deg}C>\frac{1}{\sqrt{r}}\sum _{i=1}^r{m}_i.$

The condition Template:Math is necessary: The cases Template:Math and Template:Math are distinguished by whether or not the anti-canonical bundle on the blowup of Template:Math at a collection of Template:Mvar points is nef. In the case where Template:Math, 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 Template:Mvar 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.

References

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