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.

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 $k [x 1, ..., x n]$ 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.

Statement

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

in

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 condition $r > 9$ is 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. In the case where $r \leq 9$ , 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

Harbourne, Brian (2001), "On Nagata's conjecture", Journal of Algebra, 236 (2): 692–702, arXiv:math/9909093, doi:10.1006/jabr.2000.8515, MR 1813496.
Nagata, Masayoshi (1959), "On the 14-th problem of Hilbert", American Journal of Mathematics, 81 (3): 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, hdl:10525/1746, MR 2098342.