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I have the following question?

Let $f:X\rightarrow Y$ be surjective projective morphism between smooth projective variety. I learned that if $dimY=1$, 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).

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).

I have the following question?

Let $f:X\rightarrow Y$ be surjective projective morphism between smooth projective variety. I learned that if $dimY=1$, 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).

Let $f:X\rightarrow Y$ be surjective projective morphism between projective variety ($Y$ can be singular).

Q2 Is it true that the direct image $f_*\mathcal O_X$ is always torsion free? (I feel it's true because the product of 2 nonzero holomorphic function is nonzero?)

I have the following question?

Let $f:X\rightarrow Y$ be surjective projective morphism between smooth projective variety. I learned that if $dimY=1$, 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).

I have the following question?

Let $f:X\rightarrow Y$ be surjective projective morphism between smooth projective variety. I learned that if $dimY=1$, 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).

Let $f:X\rightarrow Y$ be projective morphism between projective variety ($Y$ can be singular).

Q2 Is it true that the direct image $f_*\mathcal O_X$ is always torsion free? (I feel it's true because the product of 2 nonzero holomorphic function is nonzero?)

I have the following question?

Let $f:X\rightarrow Y$ be surjective projective morphism between smooth projective variety. I learned that if $dimY=1$, 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).

Let $f:X\rightarrow Y$ be surjective projective morphism between projective variety ($Y$ can be singular).

Q2 Is it true that the direct image $f_*\mathcal O_X$ is always torsion free? (I feel it's true because the product of 2 nonzero holomorphic function is nonzero?)