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Nagata's conjecture on curves

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For the conjecture about wild automorphisms, see Nagata's conjecture.

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 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 p1, ..., pr are very general points in P and that m1, ..., mr are given positive integers. Then for r > 9 any curve C in P that passes through each of the points pi with multiplicity mi must satisfy
deg ⁡ C > 1 r ∑ i = 1 r m i . {\displaystyle \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 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.

References

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