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Is a finite etale cover of a product of curves again a product of curves?

The answer is no in general. Here's one way to construct an example. Take the product $A$ of two elliptic curves and an isogeny $J\to A$ with $J$ the Jacobian of some genus two curve.

How can we get easy counterexamples like this in the case of the product of two higher genus curves?

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Let $C_i \rightarrow C_i/G = C'_i$ ($i=1,2$) be finite unramified covers of degree $|G|>1$ of curves $C'_i$ each of genus at least $2$. Then the quotient of $C_1 \times C_2$ by the diagonal action of $G$ is an unramified cover of $C'_1 \times C'_2$.

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    $\begingroup$ For completeness, one needs to show that the unramified cover is not itself a product of curves. If, for instance, $C_1'$ and $C'_2$ have the same genus and $|G|=2$, this can be seen by computing $h^1(\mathcal O)$ and $h^2(\mathcal O)$. $\endgroup$
    – rita
    Commented Nov 6, 2013 at 22:11

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