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

Koszul property of projections of the Veronese cubic surface