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

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

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.