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–Biran conjecture, 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 ...

and Paul Biran, is a generalisation of Nagata's conjecture on curves to arbitrary polarised surfaces.

Statement

Let ''X'' be a smooth

algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...

and ''L'' be an

ample line bundle In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of ...

on ''X'' of degree ''d''. The Nagata–Biran conjecture states that for sufficiently large ''r'' the Seshadri constant satisfies :

\varepsilon(p_1,\ldots,p_r;X,L) = .

References

*. *. See in particular page 3 of the pdf. Algebraic surfaces Conjectures {{algebraic-geometry-stub