Nagata's conjecture on curves, named after the Japanese mathematician 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 formulated the conjecture in 1959, developing it through his research on Hilbert's fourteenth problem; the same paper also presented a counterexample to that earlier problem.
Facts
StatementFor r greater than 9 very general points in the projective plane with given positive multiplicities, any curve passing through each point with its multiplicity must have degree greater than 1 over the square root of r times the sum of the multiplicities. 1 Proposed Year Progress Toward ResolutionThe only case in which Nagata's conjecture on curves is known to hold is when r is a perfect square, which Nagata himself proved. 1 Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Nagata's Conjecture on Curves, Wikipedia
Statement section
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 > 1/sqrt(r) times the sum of mi.
History section
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.
Current status section
The only case when this is known to hold is when r is a perfect square, which was proved by Nagata.
- Lead section
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