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.

Nagata–Biran conjecture

Field	Algebraic geometry
Conjectured by	Paul Biran
Conjectured in	1999
Open problem	Yes
Consequences	Nagata's conjecture on curves

Statement

Let X be a smooth algebraic surface and L be an ample line bundle on X of degree d. The Nagata–Biran conjecture states that for sufficiently large r the Seshadri constant satisfies

${\displaystyle \varepsilon (p_{1},\ldots ,p_{r};X,L)={d \over {\sqrt {r}}}.}$

References

• Biran, Paul (1999), "A stability property of symplectic packing", Inventiones Mathematicae, 1 (1): 123–135, Bibcode:1999InMat.136..123B, doi:10.1007/s002220050306.
• Syzdek, Wioletta (2007), "Submaximal Riemann-Roch expected curves and symplectic packing", Annales Academiae Paedagogicae Cracoviensis, 6: 101–122, MR 2370584. See in particular page 3 of the pdf.