The pinched Veronese is Koszul

Original · EN

In this paper we prove that the coordinate ring of the pinched Veronese (i.e k[X³,X²Y,XY²,Y³,X²Z,Y²Z,XZ²,YZ²,Z³]⊂ k[X,Y,Z]) is Koszul. The strategy of the proof is the following: we can consider a presentation S/I where S=k[X₁,...,X₉]. Using a distinguished weight ω, it's enough to show that S/inωI is Koszul. We write inωI 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.