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.

Definition
Let $${ X }$$ be a smooth projective variety, $${ L }$$ an ample line bundle on it, $${ x }$$ a point of $${ X }$$, $${ \mathcal{C}_x }$$ = { all irreducible curves passing through $${ x }$$ }. $${ \epsilon ( L, x ) := \underset{ C \in \mathcal{C}_x } { \inf } \frac{ L \cdot C } { \operatorname{mult}_x( C ) } }$$.

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

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