The Koszul property for spaces of quadrics of codimension three

• Main Author: D'Alì, Alessio
• Format: Journal Article
• Language: English
• Published: 30-05-2016
• Subjects: Mathematics - Commutative Algebra
• DOI: 10.48550/arxiv.1605.09145

Journal of Algebra 490 (2017), pp. 256-282 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.