Math Conjectures

Nagata Conjecture

OPENMajorConjectureProposed 1959 · Full conjecture

Canonical statement

Let r≥10r\ge10, let p1,…,prp_1,\ldots,p_r be very general points of PC2\mathbb P^2_\mathbb C, meaning outside a countable union of proper Zariski-closed subsets of the configuration space, and let m1,…,mr≥0m_1,\ldots,m_r\ge0. If a plane curve of degree dd has multiplicity at least mim_i at every pip_i, then
dr>∑i=1rmi. d\sqrt r>\sum_{i=1}^{r}m_i.
View source LaTeX
Let \(r\ge10\), let \(p_1,\ldots,p_r\) be very general points of \(\mathbb P^2_\mathbb C\), meaning outside a countable union of proper Zariski-closed subsets of the configuration space, and let \(m_1,\ldots,m_r\ge0\). If a plane curve of degree \(d\) has multiplicity at least \(m_i\) at every \(p_i\), then
\[
  d\sqrt r>\sum_{i=1}^{r}m_i.
\]
AG-SURF-009:[Nagata1959Rational][CilibertoMiranda2001][HarbourneRoé2003Nagata]
Algebraic geometryAlgebra and representation theory

Notes

The Nagata conjecture concerns how much vanishing one can impose on plane curves at points in general position: for r≥10r\ge10 very general points p1,…,pr∈P2p_1,\dots,p_r\in\mathbb P^2 and prescribed multiplicities mim_i, every curve of degree dd with multiplicity at least mim_i at each pip_i must satisfy dr>∑imid\sqrt r>\sum_i m_i. Nagata arrived at the statement in 1959 in the course of his celebrated negative solution of Hilbert's fourteenth problem [Nagata1959Rational].

Nagata himself proved the conjecture whenever rr is a perfect square [Nagata1959Rational]. For general nonsquare rr it remains open, beginning with the first case r=10r=10. Substantial technique has grown up around the problem: degeneration methods for planar linear systems [CilibertoMiranda2001] and detailed studies of linear systems with multiple base points [HarbourneRoé2003Nagata] give partial information in various regimes, but none reaches the very general nonsquare case.

The conjecture is still open; a resolution requires ruling out unexpected curves of low degree and high multiplicity through very general configurations of nonsquare size, already for ten points.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.