Veronesean almost binomial almost complete intersections

Abstract

The second Veronese ideal I$_{n}$ contains a natural complete intersection J$_{n}$ of the same height, generated by the principal 2-minors of a symmetric (n x 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 a direct link in the sense of complete intersection liaison. 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$_{n}$+ 〈f〉 for an explicit polynomial f constructed from the fibers of the Veronese grading map.