  Motivated by the classical Theorems of Picard and Siegel and their
generalizations, we define the notion of an {\it essentially large} effective
divisor and derive some of its geometric and arithmetic consequences. We then
prove that on a nonsingular projective variety $X$ whose codimension is no
greater than $\dim X-2$, every effective divisor with $\dim X +2$ or more
components in general position is essentially large.
