A_6 invariant curves of genera 10 and 19

We study smooth curves on which the alternating group $\mathfrak{A}_{6}$ acts faithfully. Let $\mathcal{V} \subset PGL(3, \mathbb{C})$ be the Valentiner group, which is isomorphic to $\mathfrak{A}_{6}$. We see that there are integral $\mathcal{V}$-invariant curves of degree $12$ which have geometric genera $10$ and $19$. On the other hand, if $\mathfrak{A}_{6}$ acts faithfully on a curve $C$ of genus $10$ or $19$, then we give an explicit description of the extension $k(C / \mathfrak{A}_{5}) \rightarrow k(C / \mathfrak{A}_{6})$ for any icosahedral subgroup $\mathfrak{A}_{5}$. Using this, we show the uniqueness of smooth projective curves of genera $10$ and $19$ whose automorphism groups contain $\mathfrak{A}_{6}$.