Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties
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 <i>n</i>-tuple of collinear points can be li...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-09-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/12/19/3041 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850184962563637248 |
|---|---|
| author | Oliver Clarke Giacomo Masiero Fatemeh Mohammadi |
| author_facet | Oliver Clarke Giacomo Masiero Fatemeh Mohammadi |
| author_sort | Oliver Clarke |
| collection | DOAJ |
| description | 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 <i>n</i>-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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mo>×</mo><mn>4</mn></mrow></semantics></math></inline-formula> 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. |
| format | Article |
| id | doaj-art-e52fe1e4ad7549dd973eb15efed2376b |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-e52fe1e4ad7549dd973eb15efed2376b2025-08-20T02:16:54ZengMDPI AGMathematics2227-73902024-09-011219304110.3390/math12193041Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit VarietiesOliver Clarke0Giacomo Masiero1Fatemeh Mohammadi2School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, UKDepartment of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, BelgiumDepartment of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, BelgiumWe 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 <i>n</i>-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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mo>×</mo><mn>4</mn></mrow></semantics></math></inline-formula> 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.https://www.mdpi.com/2227-7390/12/19/3041matroidspoint-line configurationsmatroid varietiesGrassmann–Cayley algebra |
| spellingShingle | Oliver Clarke Giacomo Masiero Fatemeh Mohammadi Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties Mathematics matroids point-line configurations matroid varieties Grassmann–Cayley algebra |
| title | Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties |
| title_full | Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties |
| title_fullStr | Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties |
| title_full_unstemmed | Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties |
| title_short | Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties |
| title_sort | liftable point line configurations defining equations and irreducibility of associated matroid and circuit varieties |
| topic | matroids point-line configurations matroid varieties Grassmann–Cayley algebra |
| url | https://www.mdpi.com/2227-7390/12/19/3041 |
| work_keys_str_mv | AT oliverclarke liftablepointlineconfigurationsdefiningequationsandirreducibilityofassociatedmatroidandcircuitvarieties AT giacomomasiero liftablepointlineconfigurationsdefiningequationsandirreducibilityofassociatedmatroidandcircuitvarieties AT fatemehmohammadi liftablepointlineconfigurationsdefiningequationsandirreducibilityofassociatedmatroidandcircuitvarieties |