Veronesean almost binomial almost complete intersections

TL;DR

This paper investigates the algebraic structure of the second Veronese ideal, revealing its connections to complete intersections, liaison theory, and binomial ideals, with explicit descriptions for even dimensions.

Contribution

It characterizes subintersections of the primary decomposition of a natural complete intersection within the Veronese ideal, linking to liaison theory and providing explicit polynomial descriptions.

Findings

For even n, the Veronese ideal is Gorenstein.

Explicit polynomial f describes the intersection of primary components.

Connections to liaison theory and binomial ideals are established.

Abstract

The second Veronese ideal $I_n$ contains a natural complete intersection $J_n$ generated by the principal $2$-minors of a symmetric $(n\times n)$-matrix. We determine subintersections of the primary decomposition of $J_n$ where one intersectand is omitted. If $I_n$ is omitted, the result is the other end of a complete intersection link as in liaison theory. These subintersections also yield interesting insights into binomial ideals and multigraded algebra. For example, if $n$ is even, $I_n$ is a Gorenstein ideal and the intersection of the remaining primary components of $J_n$ equals $J_n+\langle f \rangle$ for an explicit polynomial $f$ constructed from the fibers of the Veronese grading map.