In this paper, we study
Weierstrass points P on smooth curves with two prescribed non-gaps n and s such
that s = en + d with 0 < d < n. Let ℳ be a fine moduli space of smooth curves of
genus g (with some extra structure) and let p :X→ℳ be the associated universal
family. Let Wn,s= {x ∈ X : n is the first non-gap of x and dim(|sx|) ≥ e + 1}. Let Z
be an irreducible component of Wn,s and assume that |sx| is a simple linear system
on p−1(p(x)) if x is a general point on Z. We prove that dim(Z) = n + s + g − 4 −e
and dim(|sx|) = e + 1. We give an example which shows that we cannot omit the
assumption “|sx| is a simple linear system”. We prove that such a component Z exists
if and only if e(n − 1) + d ≤ g ≤ ((n − 1)(s − 1) + 1 − (n,s))∕2. Finally, we derive
some existence results of Weierstrass points.
