A rational singularity is a singularity of a
complex variety $X$ such that for any
resolution $\pi:\; \tilde X\rightarrow X$ the
higher direct images $R^i\pi\_\*(O\_{\tilde X})$
vanish for all $i>0$. Suppose that $X$
has isolated rational singularity,
and $\tilde X\rightarrow X$ its resolution.
I expect that the fiber of $\pi$ over
the singular point is rationally
connected; I would be very grateful
for any reference to this. I need to apply
this to the local situation, so it would be
especially nice if the argument does not
use projectivity.



Does a resolution of a rational singularity have rationally connected fibers?
No. For instance the cone over an Enriques surface (with respect to any projective embedding) has rational singularity, but Enriques surface is not rationally connected.


