The Koszul property for spaces of quadrics of codimension three

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 Gröbner basis of quadrics even after a change of coordinates, thus settling in the negative a question asked by Conca.

26 pages. Comments are very welcome!

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