  We study the question of when a ring can be realized as a direct summand of a
regular ring by examining the case of homogeneous coordinate rings. We present
very strong obstacles to expressing a graded ring with isolated singularity as
a finite graded direct summand. For several classes of examples (del Pezzo
surfaces, hypersurfaces), we give a complete classification of which coordinate
rings can be expressed as direct summands (not necessarily finite), and in
doing so answer a question of Hara about the FFRT property of the quintic del
Pezzo. We also examine what happens in the case where the ring does not have
isolated singularities, through topological arguments: as an example, we give a
classification of which coordinate rings of singular cubic surfaces can be
written as finite direct summands of regular rings.
