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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.