The pinched Veronese is Koszul

TL;DR

This paper proves that the coordinate ring of the pinched Veronese variety is Koszul by analyzing a specific ideal's initial ideal and employing an extended Koszul filtration concept.

Contribution

It introduces an extended Koszul filtration method and applies it to prove the Koszul property of the pinched Veronese coordinate ring.

Findings

The coordinate ring of the pinched Veronese is Koszul.

A new approach using extended Koszul filtrations is effective.

The initial ideal analysis simplifies the proof of Koszulness.

Abstract

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 $S/I.$