  In math.AG/0407496, a new construction of limit linear series is presented
which functorializes and compactifies the original construction of Eisenbud and
Harris, using a new space called the linked Grassmannian. The boundary of the
compactification consists of crude limit series, and maps with
positive-dimensional fibers to crude limit series of Eisenbud and Harris. In
this paper, we carry out a careful analysis of the linked Grassmannian to
obtain an upper bound on the dimension of the fibers of the map on crude limit
series, thereby concluding an upper bound on the dimension of the locus of
crude limit series, and obtaining a simple proof of the Brill-Noether theorem
using only the limit linear series machinery. We also see that on a general
reducible curve, even crude limit series may be smoothed to nearby fibers.
