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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.

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

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