Let $f\colon X\to Y$ be a morphism of schemes. I am interested in the following property (too long for the title):
$$ f_{*}\mathcal{O}_{X}=\mathcal{O}_{Y} \quad \text{(P)}$$
Under very reasonable assumptions ($f$ projective between noetherian schemes), this implies that the fibers are connected (see Corollary III.11.3 in Hartshorne's Algebraic Geometry). Under still reasonable assumptions ($f$ proper over a field of characteristic zero with $X$ noetherian and $Y$ noetherian and normal), having connected fibers implies (P) (see Section 1.13 in Debarre's Higher-Dimensional Algebraic Geometry). Hence the title.
What can we say about morphisms with property (P) in general? Does (P) have any nice stability properties? Is there a name for them?
I could not find much in the literature. Do we need some extra assumptions to make property (P) interesting?
For example, in Lazarsfeld's Positivity in Algebraic Geometry I, he defines an algebraic fibre space as a surjective projective mapping $f\colon X\to Y$ between varieties (reduced and irreducible) with property (P). Then he lists some nice properties of algebraic fibre spaces. This makes me think that property (P) alone may not be that interesting.
[P.s. in the previous definition of algebraic fibre space, doesn't surjective follow from projective and (P)? If the image of $X$ is a proper closed subset of $Y$, then $\mathcal{O}_{Y}$ would have no global sections over the dense open subset $Y\setminus f(X)$, right?]
