Koszul property of projections of the Veronese cubic surface

TL;DR

This paper classifies projections of the Veronese cubic surface with Koszul coordinate rings and provides a theoretical proof of the Koszulness of the pinched Veronese, extending previous results on diagonal algebras.

Contribution

It offers a purely theoretical proof of the Koszul property for the pinched Veronese, expanding understanding of homological properties of certain algebraic projections.

Findings

Classified Koszul projections of the Veronese cubic surface

Provided a theoretical proof for the Koszulness of the pinched Veronese

Extended results on homological properties of diagonal algebras to complete intersections

Abstract

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.