  The dimension of the moduli space of smooth pointed curves with prescribed
Weierstrass semigroup at the market point is computed for three families of
symmetric semigroups of multiplicity six. We also collect the dimensions of
such moduli spaces for all semigroups of genus not greater than seven. A
question related to an improvement of a Deligne--Pinkham's bound is also
formulated, suggesting that the positive graded part of the first module of the
cotangent complex associated to a semigroup algebra is a missing invariant. The
answer for this question is positive for all these moduli that we know their
dimensions.
