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Let $X$ be a smooth compact projective manifold and $\pi:Y\rightarrow X$ a Galois covering map. Is it always possible to pull back a family of curves on $X$ to $Y$?

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How do you mean? Pull them back one by one. –  Berci Feb 21 '13 at 12:41
Is this question essentially "If I have a nice map, does pulling back by this map preserve maps of a particular other nice type?" –  Aaron Feb 21 '13 at 21:28
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