Varieties in Cages: a Little Zoo of Algebraic Geometry

Abstract

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.

arXiv comment17 pages, 5 figures