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

$\epsilon (p_1,\dots ,p_r;X,L)=\frac{d}{\sqrt{r}}.$

References

• Biran, Paul (1999), "A stability property of symplectic packing", Inventiones Mathematicae 1 (1): 123–135, doi:10.1007/s002220050306, Bibcode: 1999InMat.136..123B .
• Syzdek, Wioletta (2007), "Submaximal Riemann-Roch expected curves and symplectic packing", Annales Academiae Paedagogicae Cracoviensis 6: 101–122, http://studmath.up.krakow.pl/index.php/studmath/article/viewFile/48/41 . See in particular page 3 of the pdf.