A_6 invariant curves of genera 10 and 19

TL;DR

This paper investigates smooth algebraic curves with faithful actions of the alternating group A6, classifies invariant curves of degrees 12 with genera 10 and 19, and establishes the uniqueness of such curves with automorphism group containing A6.

Contribution

It provides explicit descriptions of A6-invariant curves of degrees 12 with genera 10 and 19, and proves the uniqueness of these curves among those with automorphism group containing A6.

Findings

Existence of integral A6-invariant curves of degree 12 with genera 10 and 19.

Explicit description of the extension of function fields related to A5 and A6 actions.

Uniqueness of smooth projective curves of genera 10 and 19 with automorphism groups containing A6.

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 $k(C / \mathfrak{A}_{5}) \rightarrow k(C / \mathfrak{A}_{6})$ 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 $\mathfrak{A}_{6}$.