Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

A Representation Theorem for Generic Line Arrangements in the Plane


Affiliations
1 Center for Study of Science, Technology and Policy, # 18 & #19, 10th Cross, Mayura Street, Papanna Layout, Nagashettyhalli, RMV II Stage, Bengaluru - 560094, India
     

   Subscribe/Renew Journal


In this article, we prove a representation theorem that any generic line arrangement in the plane over an ordered field can be represented isomorphically by a very generic line arrangement in the sense of C. A. Athanasiadis [2] with a given set of distinct slopes of the same cardinality.

Keywords

Ordered Fields, Line Arrangements in the Plane, Combinatorial Cycle Invariants, Elementary Collineation Transformation, Global Cyclicity, Concurrency Arrangement.
Subscription Login to verify subscription
User
Notifications
Font Size


  • M. Aigner and G. M. Ziegler, Proofs from THE BOOK, Springer, Berlin, 2018. viii+326 pp, ISBN-13: 978-3-662-57264-1; 978-3-662-57265-8, https://doi.org/10.1007/ 978-3-662-44205-0, MR3823190
  • C. A. Athanasiadis, The largest intersection lattice of a discriminantal arrangement, Beitr¨age zur Algebra und Geometrie, Contributions to Algebra and Geometry, Vol. 40(2) (1999), 283–289, https://www.emis.de/journals/BAG/vol.40/no.2/1.html, MR1720104
  • J. E. Goodman and R. Pollack, On the Combinatorial Classification of Non-degenerate Configurations in the Plane, Journal of Combinatorial Theory Series A. 29(2), (1980), 220–235, ISSN 0097-3165, https://doi.org/10.1016/0097-3165(80)90011-4, MR0583961
  • B. Gr¨unbaum, Arrangements and Spreads, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 10. American Mathematical Society Providence, R.I., 1972. iv+114 pp, ISBN-13: 978-0-8218-1659-2, https://bookstore.ams.org/cbms-10, MR0307027
  • B. Gr¨unbaum, Convex Polytopes, Second Edition, Graduate Texts in Mathematics, 221, Springer-Verlag, New York, 2003. xvi+468 pp. ISBN-13: 978-0-387-40409-7, https:// doi.org/10.1007/978-1-4613-0019-9, MR1976856
  • N. Jacobson, Basic Algebra I, Dover Books on Mathematics, Second Edition, 2009, ISBN13: 978-0-486-47189-1, Unabridged republication originally published by W. H. Freeman and Co., San Francisco, 1985, xviii+499 pp, MR0780184
  • N. Jacobson, Basic Algebra II, Dover Books on Mathematics, Second Edition, 2009, ISBN13: 978-0-486-47187-7, Unabridged republication originally published by W. H. Freeman and Co., San Francisco, 1989, xviii+686 pp, MR1009787
  • S. Lang, Algebra, Third Edition, Graduate Texts in Mathematics, 211, Springer-Verlag, New York, 2002, xvi+914 pp, ISBN-13: 978-0-387-95385-4, https://doi.org/10.1007/ 978-1-4613-0041-0, MR1878556
  • R. P. Stanley, An introduction to hyperplane arrangements in Geometric Combinatorics, 389–496, IAS/Park City Math. Ser., 13, American Mathematical Society, Providence, R.I., 2007, ISBN-13: 978-0-8218-3736-8, https://bookstore.ams.org/pcms-13, MR2383131

Abstract Views: 32

PDF Views: 1




  • A Representation Theorem for Generic Line Arrangements in the Plane

Abstract Views: 32  |  PDF Views: 1

Authors

C. P. Anil Kumar
Center for Study of Science, Technology and Policy, # 18 & #19, 10th Cross, Mayura Street, Papanna Layout, Nagashettyhalli, RMV II Stage, Bengaluru - 560094, India

Abstract


In this article, we prove a representation theorem that any generic line arrangement in the plane over an ordered field can be represented isomorphically by a very generic line arrangement in the sense of C. A. Athanasiadis [2] with a given set of distinct slopes of the same cardinality.

Keywords


Ordered Fields, Line Arrangements in the Plane, Combinatorial Cycle Invariants, Elementary Collineation Transformation, Global Cyclicity, Concurrency Arrangement.

References





DOI: https://doi.org/10.18311/jims%2F2020%2F24873