Curves in Cages: An Algebro-Geometric Zoo

Abstract

This paper is concerned with families of plane algebraic curves that contain a given and quite special set of points X . We focus on the case in which the set X is formed by transversally intersecting pairs of lines selected from two given finite families. The union of all lines from both families is called a cage (this notion of cage will be made more precise later), and the intersection X consists of points at which a line from the first family intersects a line from the second. The points of X are called the nodes of the cage. This is a particular case of a more general problem. Let X be the intersection set of two plane algebraic curves D and E that do not share a common component, that is, do not contain a common irreducible curve. If d and e denote the degrees of D and E , respectively, then X consists of at most d · e points. When the cardinality of X is exactly d · e, X is called a complete intersection (complete intersections have many nice properties). How does one describe polynomials of degree at most k that vanish on a complete intersection X or on its subsets? This problem has a glorious history (see [1], [2], or [3]) and its generalizations are a subject of active and exciting research [4], [5], [6], [7], [8], [9]. When k is much larger then d · e, points of X impose independent restrictions on polynomials of degree k. However, for small k these restrictions fail to be independent. In such cases, we can look for maximal subsets of X that impose independent constraints. In this article, we deal with the special case of this classical problem suggested earlier, namely, the case in which both plane curves D and E are simply unions of lines and the union D ∪ E is the (d× e)-cage in question. Note that D is the zero set of a product of d linear polynomials, while E is the zero set of a product of e linear polynomials. Hence, the degrees of D and E are d and e, respectively. In order to simplify our terminology, we color the lines from D red and the ones from E blue.