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

