  Let V be the Veronese cubic surface in P^9. We classify the projections of V
to P^8 whose coordinate rings are Koszul. In particular we obtain a purely
theoretical proof of the Koszulness of the pinched Veronese, a result obtained
originally by Caviglia using filtrations, deformations and computer assisted
computations. To this purpose we extend, to certain complete intersections,
results of Conca, Herzog, Trung and Valla concerning homological properties of
diagonal algebras.
