Varieties in Cages: a Little Zoo of Algebraic Geometry

TL;DR

This paper investigates the combinatorial structure of special hyperplane arrangements called cages in projective space and characterizes the varieties containing their nodes, showing such varieties are complete intersections.

Contribution

It establishes conditions under which varieties containing certain node sets of cages must be complete intersections, revealing new geometric constraints.

Findings

Varieties containing nodes of cages are complete intersections.

Nodes impose independent conditions on varieties of bounded degree.

Special node configurations determine the structure of containing varieties.

Abstract

A $d^{\{n\}}$-cage $\mathsf K$ is the union of $n$ groups of hyperplanes in $\Bbb P^n$, 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 $X \subset \Bbb P^n$ containing them. We prove that if $X$, given by homogeneous polynomials of degrees $\leq d$, 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.