Cactus, Pascal, and Pappus Point-Line Configurations: An Algebraic-Geometric Perspective

TL;DR

This paper investigates algebraic and geometric properties of specific point-line configurations, deriving explicit equations and decompositions for their associated matroid and circuit varieties using advanced algebraic geometry techniques.

Contribution

It provides explicit generators for the matroid ideals of cactus, Pascal, and Pappus configurations and develops irreducible decompositions for their circuit varieties.

Findings

Matroid ideals for cactus, Pascal, and Pappus configurations are generated by circuit, Grassmann-Cayley, and lifting polynomials.

Irreducible decompositions are obtained for the circuit varieties of these configurations.

A shorter decomposition method is developed for configurations with points on at most two lines.

Abstract

We study point-line configurations and their associated matroid and circuit varieties. We aim to find a finite set of defining equations for matroid varieties and an irreducible decomposition for circuit varieties. To solve the former problem, we use some classical techniques from algebraic geometry, including the Grassmann-Cayley algebra and the liftability technique. From this, we can respectively derive the Grassmann-Cayley ideal, introduced by Sidman, Traves and Wheeler, and the lifting ideal, introduced by Liwski, Mohammadi, Clarke and Masiero. Since the circuit ideal, the Grassmann-Cayley ideal and the lifting ideal are contained in the matroid ideal and explicit generators are known for them, it is a natural question to identify point-line configurations for which a generating set of the matroid ideal is formed by the circuit polynomials, Grassmann-Cayley polynomials and lifting polynomials. For these point-line configurations, we obtain an explicit and finite description of the matroid variety. In this thesis, we prove that the matroid ideal of cactus configurations, the Pascal configuration and the Pappus configuration can be generated by these three types of polynomials. To find an irreducible decomposition for the circuit varieties of point-line configurations, we use the decomposition strategy developed by Clarke, Grace, Mohammadi and Motwani. If the point-line configuration has some points lying on at most two lines, we develop a shorter alternative as well. We find such a decomposition for cactus configurations, up to irredundancy. Moreover, we find an irreducible decomposition for the Pascal configuration and the third configuration $9_3$, which is a point-line configuration with nine points and nine lines, such that every point is on three lines and every line contains three points.