Nagata's conjecture on curves
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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. 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:
- 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
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.