This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Open AccessFeature PaperArticle
Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties
by
Oliver Clarke
Oliver Clarke 1,
Giacomo Masiero
Giacomo Masiero 2 and
Fatemeh Mohammadi
Fatemeh Mohammadi 2,3,4,*
1
School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, UK
2
Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium
3
Department of Computer Science, KU Leuven, Celestijnenlaan 200A, B-3001 Leuven, Belgium
4
Department of Mathematics and Statistics, UiT—The Arctic University of Norway, 9037 Tromsø, Norway
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(19), 3041; https://doi.org/10.3390/math12193041 (registering DOI)
Submission received: 22 August 2024
/
Revised: 21 September 2024
/
Accepted: 25 September 2024
/
Published: 28 September 2024
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 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.
Share and Cite
MDPI and ACS Style
Clarke, O.; Masiero, G.; Mohammadi, F.
Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties. Mathematics 2024, 12, 3041.
https://doi.org/10.3390/math12193041
AMA Style
Clarke O, Masiero G, Mohammadi F.
Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties. Mathematics. 2024; 12(19):3041.
https://doi.org/10.3390/math12193041
Chicago/Turabian Style
Clarke, Oliver, Giacomo Masiero, and Fatemeh Mohammadi.
2024. "Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties" Mathematics 12, no. 19: 3041.
https://doi.org/10.3390/math12193041
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
here.
Article Metrics
Article Access Statistics
For more information on the journal statistics, click
here.
Multiple requests from the same IP address are counted as one view.