  We study the properties of F-rationality and F-regularity in multigraded
rings and their diagonal subalgebras. The main focus is on diagonal subalgebras
of bigraded rings: these constitute an interesting class of rings since they
arise naturally as homogeneous coordinate rings of blow-ups of projective
varieties.
  As a consequence of some of the results obtained here, it is shown that there
exist standard bigraded hypersurfaces whose rings of invariants under torus
actions have rational singularities, but are not of F-regular type. Another
application is the construction of families of rings with divisor class groups
that are finitely generated, but not discrete in the sense of Danilov.
