I know the following result is well known to the experts, but what are the good references or proofs:


Let $X,Y$ be a smooth variety with projective, birational morphism $f: X \to Y$. Let $L$ be a line bundle on $Y$, then we have the property that the higher direct image $R^if\_\*f^\* L = 0$ when $i > 0$.


This is equivalent to the statement that nonsingular variety is a rational singular variety.



Nonsingular is rational singular
I think there are two issues here:


1) The "projection formula" tells you that $R^if\_\*(f^\*L)$ is isomorphic to $R^if\_\*\mathcal{O}\_X\otimes L$: see EGA chap. 0 , Prop. 12.2.3 (in EGA III.1).


2) Then the vanishing of $R^if\_\*\mathcal{O}\_X$ for $i>0$. This is one of the many consequences of Hironaka's resolution of singularities: Resolution of singularities of an algebraic variety over a field of characteristic zero, I. Ann. of Math. (2) 79 (1964), 109–203 (the result you need is on p. 144-145).


