Unramified covers of Galois covers of low genus curves

Let $X \to Y$ be a Galois covering of curves, where the genus of $X$ is $\ge 2$ and the genus of $Y$ is $\le 2$. We prove that under certain hypotheses, $X$ has an unramified cover that dominates a hyperelliptic curve; our results apply, for instance, to all tamely superelliptic curves. Combining this with a theorem of Bogomolov and Tschinkel shows that $X$ has an unramified cover that dominates $y^2=x^6-1$, if $\Char k$ is not $2$ or $3$.

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Published 1 January 2005