  In this paper, we compute the genus of the variety of linear series of rank
$r$ and degree $d$ on a general curve of genus $g$, with ramification at least
$\alpha$ and $\beta$ at two given points, when that variety is 1-dimensional.
Our proof uses degenerations and limit linear series along with an analysis of
random staircase paths in Young tableaux, and produces an explicit
scheme-theoretic description of the limit linear series of fixed rank and
degree on a generic chain of elliptic curves when that scheme is itself a
curve.
