The pinched Veronese is Koszul
  In this paper we prove that the coordinate ring of the pinched Veronese (i.e $k[X^3,X^2Y,XY^2,Y^3,X^2Z,Y^2Z,XZ^2,YZ^2,Z^3]\subset k[X,Y,Z]$) 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 $S/in_{\omega}I$ is Koszul. We write $in_{\omega}I$ as $J+H$ where $J$ is generated by a Gr\"obner 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 $S/I.$ 
Giulio Caviglia