$\mathfrak {A}_{6}$-invariant curves of genera 10 and 19

Abstract

We study smooth curves on which the alternating group $\mathfrak {A}_{6}$ acts faithfully. Let $\mathcal {V} \subset {\text {PGL}}(3, \mathbb {C})$ 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 $k(C / \mathfrak {A}_{5}) / k(C / \mathfrak {A}_{6})$ 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 $\mathfrak {A}_{6}$.