Let $\pi:X\rightarrow Y$ be a double cover where $X$ and $Y$ are projective smooth curves.


Is it true that $R^1\pi\_\*\mathcal O\_X=0$ ? Why ? 


Thanks in advance.



Direct image of structural sheaf
X and Y are proper and irreducible. As a double cover does not crush X to a point, it must be surjective. The fibres are then finite sets which implies that it is a finite morphism. Finite morphisms are affine. Affine morphisms do not have higher direct image (this is just the relative version of the theorem which says that affine schemes do not have higher cohomology with coefficients in quasi-coherent sheaves).


