mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Nagata conjecture on curves, named after

Masayoshi Nagata Masayoshi Nagata (Japanese: 永田雅宜 ''Nagata Masayoshi''; February 9, 1927 – August 27, 2008) was a Japanese mathematician, known for his work in the field of commutative algebra. Work Nagata's compactification theorem shows that var ...

, governs the minimal degree required for a plane algebraic curve to pass through a collection of very general points with prescribed

multiplicities In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multip ...

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 over some field 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 are very general points in and that are given positive integers. Then for any curve in that passes through each of the points with multiplicity must satisfy ::

\deg C > \frac\sum_^r m_i.

The condition is necessary: The cases and are distinguished by whether or not the anti-canonical bundle on the blowup of at a collection of points is

nef Nef or NEF may refer to: Businesses and organizations * National Energy Foundation, a British charity * National Enrichment Facility, an American uranium enrichment plant * New Economics Foundation, a British think-tank * Near East Foundation, ...

. In the case where , the cone theorem essentially gives a complete description of the

cone of curves In mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an algebraic variety X is a combinatorial invariant of importance to the birational geometry of X. Definition Let X be a proper variety. By definition, a (real) ''1-cycle'' ...

of the blow-up of the plane.

Current status

The only case when this is known to hold is when is a perfect square, which was proved by

Nagata Nagata is a surname which can be either of Japanese (written: 永田 or 長田) or Fijian origin. Notable people with the surname include: *Akira Nagata (born 1985), Japanese vocalist and actor * Alipate Nagata, Fijian politician *Anna Nagata (bor ...

. Despite much interest, the other cases remain open. A more modern formulation of this conjecture is often given in terms of

Seshadri constant In algebraic geometry, a Seshadri constant is an invariant of an ample line bundle ''L'' at a point ''P'' on an algebraic variety. It was introduced by Jean-Pierre Demailly, Demailly to measure a certain ''rate of growth'', of the tensor powers of ' ...

s and has been generalised to other surfaces under the name of the

Nagata–Biran conjecture In mathematics, the Nagata–Biran conjecture, named after Masayoshi Nagata and Paul Biran, is a generalisation of Nagata's conjecture on curves to arbitrary polarised surfaces. Statement Let ''X'' be a smooth algebraic surface and ''L'' be an am ...

References

*. *. *. {{DEFAULTSORT:Nagata's Conjecture On Curves Algebraic curves Conjectures