The pinched Veronese is Koszul

Abstract

In this paper we prove that the coordinate ring of the pinched Veronese (i.e $k[X^3,X^{2}Y,X{Y}^2,Y^3,X^{2}Z,Y^{2}Z,X{Z}^2,Y{Z}^2,Z^3]\subset k[X,Y,Z]$) is Koszul. The strategy of the proof is the following: we can consider a presentation $S/I$ where $S=k[X_1,...,X_9]$. Using a distinguished weight $\omega$, it's enough to show that $S/i{n}_{\omega }I$ is Koszul. We write $i{n}_{\omega }I$ as $J+H$ where $J$ is generated by a Gröbner basis of quadrics. Finally, we present an extension of the notion of Koszul filtration and we use it to show that $(J+H)/J$ has a linear free resolution over $S/J.$ This implies the Koszulness of $S/I.$

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• Koszul algebras and Gröbner bases of quadrics