In algebraic geometry, a Seshadri constant is an invariant of an ample line bundle at a point on an algebraic variety. It was introduced by Jean-Pierre Demailly to measure a certain "rate of growth" of the tensor powers of in terms of the jets of the sections of the . The object was the study of the Fujita conjecture. The constant's name stems from its connection to C. S. Seshadri's criterion for ampleness.
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 be a smooth projective variety, an ample line bundle on , a point of , and the set of all irreducible curves passing through . Define the Seshadri constant of at by ,
where denotes the intersection number of and and measures how many times the curve passes through .
If is an abelian variety, then is independent of the point chosen, and we write simply .
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