Signatures of alternating group actions with non-zero quotient genus

We classify up to signature all the ways the alternating group $A_n$ can act on a compact Riemann surfaces when the quotient genus is greater than $0$. In particular, we prove that for $A_n$ with $n>6$ every potential signature for the group acting with quotient genus greater than $0$ is an actual signature. We also show that in the case of $n=5$, respectively $n=6$, the only failure is for $[1;2]$, respectively $[1;3]$. Along the way we also prove that for any finite simple non-abelian group, all potential signatures with quotient genus greater than $1$ are actual signatures.