Seshadri constant

In

The name is in honour of the Indian mathematician C. S. Seshadri.

It is known that

Let ${\displaystyle {X}}$ be a smooth projective variety, ${\displaystyle {L}}$ an ample line bundle on it, ${\displaystyle {x}}$ a point of ${\displaystyle {X}}$, ${\displaystyle {{\mathcal {C}}_{x}}}$ = { all irreducible curves passing through ${\displaystyle {x}}$ }.

Here, ${\displaystyle {L\cdot C}}$ denotes the intersection number of ${\displaystyle {L}}$ and ${\displaystyle {C}}$, ${\displaystyle {\operatorname {mult} _{x}(C)}}$ measures how many times ${\displaystyle {C}}$ passing through ${\displaystyle {x}}$.

Definition: One says that ${\displaystyle {\epsilon (L,x)}}$ is the Seshadri constant of ${\displaystyle {L}}$ at the point ${\displaystyle {x}}$, a real number. When ${\displaystyle {X}}$ is an abelian variety, it can be shown that ${\displaystyle {\epsilon (L,x)}}$ is independent of the point chosen, and it is written simply ${\displaystyle {\epsilon (L)}}$.

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, Maximilian (2018), On the Integrality of Seshadri Constants of Abelian Surfaces,
  arXiv:1805.05413