I am reading Vakil's algebraic geometry. There is exercise:
Show that the nonhyperelliptic curves of genus 3 form a family of dimension 6.
Then there is a hint: counting the dimension of the family of regular quartic curves, and quotient by $\mathrm{Aut} \mathbb{P}^{2}=\mathrm{PGL}(3)$. (We know that , there is a bijection between nonhyperelliptic genus $3$ curves, and plane quartic curves up to projective linear transformations)
But the dimension of the family of regular quartic curves is $\binom{6}{2}=15$, and $\mathrm{dim} \mathrm{Aut} \mathbb{P}^{2}=9-1=8$, so I think the dimension of the family should be 7 instead of 6? Is it right? If not, could you tell me where I make a mistake? Thank you very much.