Veronesean almost binomial almost complete intersections

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\mathrm{.}Wedeterminesubintersectionsoftheprimarydecompositionof$J_n$whereoneintersectandisomitted\mathrm{.}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 $J_n+⟨f⟩$ for an explicit polynomial $f$ constructed from the fibers of the Veronese grading map.