Moduli of Weierstrass points

A smooth algberaic curve of genus g has finitely many special points, called Weierstrass points, which are those points P such that the divisor gP has positive rank. I will consider the question: given a sequence of integers, what is the geometry of the space of pointed curves (C,P) such that the ranks of the divisors nP form form this sequence? In particular, I will give an invariant which is a lower bound on the dimension of a component of this space, and prove that this lower bound is attained whenever this invariant is less than g, generalizing results of Eisenbud and Harris. The argument proceeds by degeneration to chains of elliptic curves, and by considering limits of linear series on such chains.