Groebner bases for spaces of quadrics of codimension 3

TL;DR

This paper investigates the structure of Artinian graded algebras defined by quadrics with small second graded component, showing they are mostly defined by quadratic Gr"obner bases except for a specific class of complete intersections.

Contribution

It proves that such algebras are generally defined by quadratic Gr"obner bases, identifying a unique exception involving a specific complete intersection.

Findings

Most algebras are defined by quadratic Gr"obner bases.

A unique exception involves a complete intersection of three quadrics.

The results depend on the algebraically closed field of characteristic not 2.

Abstract

Let $R=\oplus_{i\geq 0} R_i$ be an Artinian standard graded $K$-algebra defined by quadrics. Assume that $\dim R_2\leq 3$ and that $K$ is algebraically closed of characteristic $\neq 2$. We show that $R$ is defined by a Gr\"obner basis of quadrics with, essentially, one exception. The exception is given by $K[x,y,z]/I$ where $I$ is a complete intersection of 3 quadrics not containing the square of a linear form.