Does anyone know any information on the question of the dimension of moduli space of pointed curves with fixed Weierstrass semigroup? Some conjecture?

If the complement of the semigroup (the gaps) is $a_1,\ldots,a_g$, then the weight is $w=\sum (a_ii)$. I think you expect codimension $w1$ (for small $w>0$) for the space of curves of genus $g$ having the given semigroup. However, this cannot work for large $w$. Have a look at the papers of Steven Diaz. 


One good reference is "Recent progress in the study of Weierstrass points" by Eisenbud and Harris in Geometry Today, Birkhauser, 1985. 

