被捏的韦罗内塞是科苏尔
The pinched Veronese is Koszul
摘要：
本文证明了Pinched Veronese（即$k[X^3,X^2Y,XY^2,Y^3,X^2Z,Y^2Z,XZ^2,YZ^2,Z^3]\subset k[X,Y,Z]$）的坐标环是Koszul.证明的策略如下：我们可以考虑一个演示文稿$S/I$，其中$S=k[X_1,...,X_9]$。使用可区分的权重$ω$，就足以表明$S/in_ωI$是Koszul。我们将$in_ωI$写成$J+H$，其中$J$由二次曲面的Gröbner基生成。最后，我们给出了Koszul滤子概念的一个推广，并利用它证明了$(J+H)/J$在$S/J.$上具有线性自由分辨。这暗示了$S/I.$的Koszul性质
英文摘要：
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 $ω$, it's enough to show that $S/in_ωI$ is Koszul. We write $in_ωI$ 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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