  Let X be a smooth projective variety of dimension n in P^r. We study the
fibers of a general linear projection pi: X --> P^{n+c}, with c > 0. When n is
small it is classical that the degree of any fiber is bounded by n/c+1, but
this fails for n >> 0. We describe a new invariant of the fiber that agrees
with the degree in many cases and is always bounded by n/c+1. This implies, for
example, that if we write a fiber as the disjoint union of schemes Y' and Y''
such that Y' is the union of the locally complete intersection components of Y,
then deg Y'+deg Y''_red <= n/c+1 and this formula can be strengthened a little
further. Our method also gives a sharp bound on the subvariety of P^r swept out
by the l-secant lines of X for any positive integer l, and we discuss a
corresponding bound for highly secant linear spaces of higher dimension. These
results extend Ziv Ran's "Dimension+2 Secant Lemma".
