Seshadri constant
In algebraic geometry, a Seshadri constant is an invariant of an ample line bundle L at a point P on an algebraic variety. It was introduced by Demailly to measure a certain rate of growth, of the tensor powers of L, in terms of the jets of the sections of the Lk. The object was the study of the Fujita conjecture.
The name is in honour of the Indian mathematician C. S. Seshadri.
It is known that Nagata's conjecture on algebraic curves is equivalent to the assertion that for more than nine general points, the Seshadri constants of the projective plane are maximal. There is a general conjecture for algebraic surfaces, the Nagata–Biran conjecture.
Statement
Let be a smooth projective variety, an ample line bundle on it, a point of , = { all irreducible curves passing through }.
.
Here, denotes the intersection number of and , measures how many times passing through .
Definition. one says that is the Seshadri constant of at the point .
- Comment on the Definition. It's easy to see that is a real number.
De facto, when is an abelian variety, it can be shown that is independent of the point chosen. Therefore, in the said situation, one can legally drop x, and writes simply .
References
- Lazarsfeld, Robert (2004), Positivity in Algebraic Geometry I - Classical Setting: Line Bundles and Linear Series, Springer-Verlag Berlin Heidelberg, pp. 269–270
- Bauer, Thomas; Grimm, Felix Fritz; Schmidt, Maximalian, On the Ingegrality of Seshadri Constants of Abelian Surfaces, arXiv:1805.05413