# genus four curve with $|3p|= \mathfrak{g}^1_3$

Let $M_4$ be the moduli space of genus four curves. Let $\Sigma \subset M_4$ be the locus such that for $X \in \Sigma$, there is a point $p$ on $X$, with $\dim H^0(X, \mathcal{O}(3p)) =2$. What is the codimension of this locus in $M_4$?

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Counting parameters suggests that $\Sigma$ is a divisor: A curve lying in this locus has a degree $3$ map to $\mathbb{P}^1$ which is totally ramified above one point. So by the Riemann-Hurwitz formula we get $2g -2 = 6 = 3(-2) + 2 + r$ where $r$ is the number of other ramification points which we assume are all simple (to get maximal dimension), so $r = 12 -2 = 10$. Thus there are 11 ramification points and since $Aut(\mathbb{P}^1)$ is three dimensional, it follows that the number of parameters is $8$. – ulrich Apr 15 '12 at 10:27