Nagata's conjecture on curves
# 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 ≤ 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 ≤ 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.

