Alternating groups as monodromy groups in positive characteristic

Let $X$ be a generic curve of genus $g$ defined over an algebraically closed field $k$ of characteristic $p\ge 0$. We show that for $n$ sufficiently large there exists a tame rational map $f:X\to {\text{PP}}_k^1$ with monodromy group $A_n$. This generalizes a result of Magaard–V"olklein to positive characteristic.