Varieties in Cages: A Little Zoo of Algebraic Geometry

A \(d^{\{n\}}\)-cage \(\mathsf K\) is the union of n groups of hyperplanes in \(\mathbb P^n\), each group containing d members. The hyperplanes from the distinct groups are in general position, thus producing \(d^n\) points where hyperplanes from all groups intersect. These points are called the nodes of \(\mathsf K\). We study the combinatorics of nodes that impose independent conditions on the varieties \(X \subset \mathbb P^n\) containing them. We prove that if X, given by homogeneous polynomials of degrees \(\le d\), contains the points from such a special set \(\mathsf A\) of nodes, then it contains all the nodes of \(\mathsf K\). Such a variety X is very special: in particular, X is a complete intersection.