Brill-Noether generality of binary curves

We show that the space G((d) under bar)(r) (X) of linear series of certainmulti-degree (d) under bar = (d(1), d(2)) (including the balanced ones) and rank r on a general genus-g binary curve X has dimension rho(g, r, d) = g - (r + 1)( g - d + r) if nonempty, where d = d(1) + d(2). This generalizes Caporaso's result from the case r <= 2 to arbitrary rank, and shows that the space of Osserman-limit linear series on a general binary curve has the expected dimension, which was known for r <= 2. In addition, we show that the space G((d) under bar)(r)(X) is still of expected dimension after imposing certain ramification conditionswith respect to a sequence of increasing effective divisors supported on two general points P-i epsilon Z(i), where i = 1, 2 and Z(1), Z(2) are the two components of X. Our result also has potential application to the lifting problem of divisors on graphs to divisors on algebraic curves.