Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties

Oliver Clarke, Giacomo Masiero and Fatemeh Mohammadi ()
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Oliver Clarke: School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, UK
Giacomo Masiero: Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium
Fatemeh Mohammadi: Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium

Mathematics, 2024, vol. 12, issue 19, 1-34

Abstract: We study point-line configurations through the lens of projective geometry and matroid theory. Our focus is on their realization spaces, where we introduce the concepts of liftable and quasi-liftable configurations, exploring cases in which an n -tuple of collinear points can be lifted to a nondegenerate realization of a point-line configuration. We show that forest configurations are liftable and characterize the realization space of liftable configurations as the solution set of certain linear systems of equations. Moreover, we study the Zariski closure of the realization spaces of liftable and quasi-liftable configurations, known as matroid varieties, and establish their irreducibility. Additionally, we compute an irreducible decomposition for their corresponding circuit varieties. Applying these liftability properties, we present a procedure to generate some of the defining equations of the associated matroid varieties. As corollaries, we provide a geometric representation for the defining equations of two specific examples: the quadrilateral set and the 3 × 4 grid. While the polynomials for the latter were previously computed using specialized algorithms tailored for this configuration, the geometric interpretation of these generators was missing. We compute a minimal generating set for the corresponding ideals.

Keywords: matroids; point-line configurations; matroid varieties; Grassmann–Cayley algebra (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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