# 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. 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}, ..., *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:

**Nagata Conjecture.**Suppose*p*_{1}, ...,*p*are very general points in_{r}**P**^{2}and that*m*_{1}, ...,*m*are given positive integers. Then for_{r}*r*> 9 any curve C in**P**^{2}that passes through each of the points*p*with multiplicity_{i}*m*must satisfy_{i}

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.

The condition *r* > 9 is easily seen to be necessary. The cases *r* > 9 and *r* ≤ 9 are distinguished by whether or not the anti-canonical bundle on the blowup of **P**^{2} at a collection of r points is nef.

## References

- Harbourne, Brian (2001), "On Nagata's conjecture",
*Journal of Algebra*,**236**(2): 692–702, doi:10.1006/jabr.2000.8515, MR 1813496. - Nagata, Masayoshi (1959), "On the 14-th problem of Hilbert",
*American Journal of Mathematics*,**81**: 766–772, doi:10.2307/2372927, JSTOR 2372927, MR 0105409. - Strycharz-Szemberg, Beata; Szemberg, Tomasz (2004), "Remarks on the Nagata conjecture",
*Serdica Mathematical Journal*,**30**(2-3): 405–430, MR 2098342.