A_6 invariant curves of genera 10 and 19

Abstract

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 $k(C/{\mathfrak{A}}_5)\to k(C/{\mathfrak{A}}_6)$ 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 ${\mathfrak{A}}_6$.

21 pages