In a comment to this question about Stein factorization, Tabes Bridges writes

Moduli technicalities (particularly in positive characteristic), the condition $f_∗\mathcal{O}_X=\mathcal{O}_Y$ basically means that the fibers of $f$ are connected.

I wonder what those technicalities actually are, especially in characteristic zero.

In the proof of Corollary III 11.4, Hartshorne actually proves the statement

Let $f: X \to Y$ be a birational projective morphism of noetherian integral schemes, and assume that $Y$ is normal. Then $f_* \mathcal{O}_X = \mathcal{O}_Y$.

Are there other criterions, where I don't need normality?

Context I have a rather concrete morphism $X \to \mathbb{P}^n$ and I want to show that its image is a smooth projective variety. So my advisor suggested to look at the theorem of formal functions, and calculate $\lim H^i(X_n, \mathcal{O}_n)$ to show that $\mathcal{O}_{Y,y}$ (resp. its completion) is regular. But the theorem on formal functions only deals with the completion of $(f_* \mathcal{O}_X)_y$, so I think that I need $\mathcal{O}_Y = f_* \mathcal{O}_X$.


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