Let $ f:X\to Y$ be a morphism of schemes over $ k$ . I am wondering about, what geometric consequences $ R^qf_*O_X=0$ for $ q\geq k$ does have. I saw vanishing of higher direct images used in some proofs of coherence of structure sheaves and also for the computation of cohomology if one in addition assumes that $ f_*O_X=O_Y$ . Also there seem to be a lot of papers out there discussing cases in which the higher direct images vanish for $ q\geq 1$ .
What are natural situations in which one wants vanishing of higher direct images and what does it tell us (geometrically)?
I know this is kind of vague but I was unable to figure it out myself. Thanks in advance!