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

: $\varepsilon(p_1,\ldots,p_r;X,L) = {d \over \sqrt{r}}.$

| last = Biran | first = Paul

| journal =Inventiones Mathematicae

| pages = 123–135

| title = A stability property of symplectic packing

| doi = 10.1007/s002220050306

| volume = 1

| year = 1999| issue = 1

| bibcode = 1999InMat.136..123B

}}.

| last = Syzdek | first = Wioletta

| journal = Annales Academiae Paedagogicae Cracoviensis

| mr = 2370584

| pages = 101–122

| title = Submaximal Riemann-Roch expected curves and symplectic packing

| url = http://studmath.up.krakow.pl/index.php/studmath/article/viewFile/48/41

| volume = 6

| year = 2007}}. See in particular page 3 of the pdf.