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 r

are very general points in

P 2

and that

m 1, ..., m r

are given positive integers. Then for

r > 9

any curve

C

P 2

that passes through each of the points

p i

with multiplicity

m i

must satisfy

\deg C>{\frac {1}{{\sqrt {r}}}}\sum _{{i=1}}^{r}m_{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 $r \leq 9$ 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

Brian Harbourne, Sandra Di Rocco, Tomasz Szemberg, and Thomas Bauer in Oberwolfach, 2010

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 .

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