# When does $\mathcal{O}_Y = f_* \mathcal{O}_X$ hold?

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 $$\mathcal{O}_Y = f_* \mathcal{O}_X$$.