Linear series on general curves with prescribed incidence conditions

Using degeneration and Schubert calculus, we consider the problem of computing the number of linear series of given degree d and dimension r on a general curve of genus g satisfying prescribed incidence conditions at n points. We determine these numbers completely for linear series of arbitrary dimension when d is sufficiently large, and for all d when either r = 1 or n = r + 2. Our formulas generalize and give new proofs of recent results of Tevelev and of Cela-Pandharipande-Schmitt.