  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 $\epsilon(L,x)$ --- which in effect measures how positive $L$
is locally near a given point $x \in X$. 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 $\epsilon(L,x) > e$ for all $x \in
X$, and if $L$ is VERY ample, then $\epsilon(L,x) \ge 1$ 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, $\epsilon(L,x) \ge (1/n) $ for all $x
\in X$ 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.
