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It is a basic question and I would be happy to be directed to some reference for it.

Let $f\colon X\to Y$ be a finite branched cover of smooth projective varieties, $M$ a line bundle on $Y$ and $L=f^\ast M$ its pullback to $X$. For simplicity, let's assume $\deg f=2$. Then $L$ is invariant by the involution defined by the double cover and one has a decomposition $H^0(L) = f^\ast H^0(M) \oplus V$ where $V$ is the eigenspace corresponding to $-1$ for the action of the involution on $H^0(L)$.

Questions: what are the sections in $V$? I guess it should be related to the vanishings along the branch locus, how? What happens for higher degrees (I guess the existence of the automorphism is then not automatic, but if we have it then how do we interpret the eigenspaces)?

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A good reference for cyclic covers, which includes your degree 2 case, is "Lectures on Vanishing theorems" by Esnault and Viehweg. Briefly, assuming characteristic different from 2, $f_*O_X$ decomposes as $O_Y\oplus R^{-1}$, where $R$ is line bundle such that $R^{\otimes 2}=O_Y(B)$, where $B$ is the branch locus. By the projection formula $V= H^0(R^{-1}\otimes M)$.

In general, for a Galois cover, you could decompose spaces/sheaves using characters of the group. But it wouldn't be as explicit.

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    $\begingroup$ For a cyclic cover of any degree the decomposition is very explicit. $\endgroup$ – Sasha Jan 22 '18 at 21:47

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