1994/08/09 by Ein, Lawrence, Küchle, Oliver, Lazarsfeld, Robert
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.alg-geom/9408003
Let L be a nef line bundle on a smooth complex projective variety X of dimension n. Demailly has introduced a very interesting invariant --- the Seshadri constant ε(L,x) --- which in effect measures how positive L is locally near a given point x ∈ X. For instance, Seshadri's criterion for ampleness may be phrased as stating that L is ample if and only if there exists a positive number e > 0 such that ε(L,x) > e for all x ∈ X, and if L is VERY ample, then ε(L,x) ≥ 1 for every x. We prove the somewhat surprising result that in each dimension n there is a uniform lower bound on the Seshadri constant of an ample line bundle L at a very general point of X. Specifically, ε(L,x) ≥ (1/n) for all x ∈ X outside the union of countably many proper subvarieties of X. Examples of Miranda show that there cannot exist a bound (independent of X and L) that holds at every point. The proof draws inspiration from two sources: first, the arguments used to prove boundedness of Fano manifolds of Picard number one; and secondly some of the geometric ideas involving zero-estimates appearing in the work of Faltings and others on Diophantine approximation and transcendence theory. We give some elementary applications of the main theorem to adjoint and pluricanonical linear series.