# Nagata's conjecture on curves

Field Algebraic geometry Masayoshi Nagata 1959 Yes r is a perfect square Nagata–Biran 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[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
$\deg C>{\frac {1}{\sqrt {r}}}\sum _{i=1}^{r}m_{i}.$ 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 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.