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

Homogeneous coordinate rings as direct summands of regular rings