mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 al ...

, governs the minimal degree required for a

plane algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

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 multipl ...

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 The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United S ...

, 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 ''Blowup'' (also styled ''Blow-Up'') is a 1966 Psychological thriller, psychological Mystery film, mystery film directed by Michelangelo Antonioni, co-written by Antonioni, Tonino Guerra and Edward Bond and produced by Carlo Ponti. It is Antoni ...

of at a collection of points is nef. 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. 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 Demailly to measure a certain ''rate of growth'', of the tensor powers of ''L'', in terms of th ...

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