Let X denote a compact
Riemann surface of genus g and suppose P ∈ X. The Weierstrass nongaps at P are
those positive integers n such that there exists a meromorphic function on X which
has a pole of order n at P and is holomorphic everywhere else. The Weierstrass
semigroup at P, which we denote by Γ(P), is the additive semigroup consisting of 0
and the nongaps. A point P is a Weierstrass point if there exists a nongap less than
g + 1 at P and is a normal Weierstrass point if Γ(P) = {0,g,g + 2,g + 3,g + 4,⋯}.
We consider here the following problem: Given a collection P1,⋯,Pn of points on X,
describe the infinitesimal variations of complex structure on X which preserve the
Weierstrass semigroups at P1,⋯,Pn. Our main result says, roughly speaking,
that normal Weierstrass points deform as independently of each other as
possible.
