The Number of Linear Series on Curves with given Ramification

We use Eisenbud and Harris’ theory of limit linear series (1986) to show that for a general smooth curve of genus g in characteristic 0, with general points Pi and indices ei such that P i(ei − 1) = 2d − 2− g, G 1 d (C, {(Pi, ei)}i) is made up of reduced points. We give a formula for the number of points, showing that it agrees with various known special cases. We also conjecture a corresponding reducedness result and formula for g d s of any dimension, and reduce this to the case of three points on P, where one need no longer consider moduli or generality.