The Koszul property for spaces of quadrics of codimension three

TL;DR

This paper proves that most quadratic standard graded algebras with a three-dimensional degree-two component over an algebraically closed field of characteristic not 2 are Koszul, with only a few exceptions, and explores properties related to Gr"obner bases.

Contribution

It classifies the Koszul property for quadratic algebras of codimension three, identifying the few non-Koszul cases and addressing a question about Gr"obner bases.

Findings

Almost all such algebras are Koszul, except for a few specific cases.

Existence of Koszul algebras without a quadratic Gr"obner basis after coordinate change.

Clarification of the structure of quadratic algebras with certain dimension conditions.

Abstract

In this paper we prove that, if $\mathbb{k}$ is an algebraically closed field of characteristic different from 2, almost all quadratic standard graded $\mathbb{k}$-algebras $R$ such that $\dim_{\mathbb{k}}R_2 = 3$ are Koszul. More precisely, up to graded $\mathbb{k}$-algebra homomorphisms and trivial fiber extensions, we find out that only two (or three, when the characteristic of $\mathbb{k}$ is 3) algebras of this kind are non-Koszul. Moreover, we show that there exist nontrivial quadratic standard graded $\mathbb{k}$-algebras with $\dim_{\mathbb{k}}R_1 = 4$, $\dim_{\mathbb{k}}R_2 = 3$ that are Koszul but do not admit a Gr\"obner basis of quadrics even after a change of coordinates, thus settling in the negative a question asked by Conca.