The slopes determined by n points in the plane

Let m 12 , m 13 , …, m n - 1 , n be the slopes of the ( n 2 ) lines connecting n points in general position in the plane. The ideal I n of all algebraic relations among the m ij defines a configuration space called the slope variety of the complete graph. We prove that I n is reduced and Cohen-Macau...

Full description

Saved in:
Bibliographic Details
Published in:Duke mathematical journal 2006-01, Vol.131 (1), p.119-165
Main Author: Martin, Jeremy L.
Format: Article
Language:English
Subjects:
05C10
13P10
14N20
57M25
Configurations and arrangements of linear subspaces
geometric and topological aspects of graph theory [See also 57M15
Gröbner bases
Janet and border bases
other bases for ideals and modules (e.g
Planar graphs
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Let m 12 , m 13 , …, m n - 1 , n be the slopes of the ( n 2 ) lines connecting n points in general position in the plane. The ideal I n of all algebraic relations among the m ij defines a configuration space called the slope variety of the complete graph. We prove that I n is reduced and Cohen-Macaulay, give an explicit Gröbner basis for it, and compute its Hilbert series combinatorially. We proceed chiefly by studying the associated Stanley-Reisner simplicial complex, which has an intricate recursive structure. In addition, we are able to answer many questions about the geometry of the slope variety by translating them into purely combinatorial problems concerning the enumeration of trees
ISSN:0012-7094
1547-7398
DOI:10.1215/S0012-7094-05-13114-3