# On the dimension of moduli space of pointed curves with fixed Weierstrass semigroup

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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For those who (like me) didn't know the terminology: "Given a curve X over a field k, an object classically studied is the Weierstrass semigroup of a k-rational point P of X. This semigroup consists of those nonnegative integers n such that there is a rational function on X with a pole of order n at P and no other poles." –  JSE Dec 7 '09 at 4:41

If the complement of the semigroup (the gaps) is $a_1,\ldots,a_g$, then the weight is $w=\sum (a_i-i)$. I think you expect codimension $w-1$ (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.