I have the following question.

Let $f:X\rightarrow Y$ be a surjective projective morphism between smooth projective varieties. I learned that if $\dim Y=1$, then $R^if_*\mathcal O_X$ is torsion free $\forall i\geq0$. I think smoothness of $X,Y$ is important here, and I wonder what about the condition $dim Y=1$?

* Q1* Is it still true that $R^if_*\mathcal O$ is torsion free $\forall i\geq0$ when $\dim Y>1$? If false, what's a natural counter example? (To show how dimension of $Y$ matters).

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