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.

## Contents

* 1 History
* 2 Statement
* 3 Current status
* 4 References

## 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[x1, ..., xn] 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 P2 and that m1, ..., mr are given positive integers. Then for r > 9 any curve C in P2 that passes through each of the points pi with multiplicity mi must satisfy

deg⁡C>1r∑i=1rmi.

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 P2 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

* "On Nagata's conjecture", Journal of Algebra 236 (2): 692–702, 2001, doi:10.1006/jabr.2000.8515 .
* Nagata, Masayoshi (1959), "On the 14-th problem of Hilbert", American Journal of Mathematics 81 (3): 766–772, doi:10.2307/2372927 .
* Strycharz-Szemberg, Beata (2004), "Remarks on the Nagata conjecture", Serdica Mathematical Journal 30 (2–3): 405–430 .

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