nLab arrangement of hyperplanes

Idea

An arrangement of hyperplanes is a finite set of hyperplanes in a (finite-dimensional) linear, affine or projective space. Usually (such as for configuration spaces of points) one studies in fact the complement of the union of the hyperplanes, and its topological and other properties. This space is the basis of many interesting fiber bundles appearing in conformal field theory, study of hypergeometric functions (Aomoto, Gelfand, Varchenko), quantum groups etc., for more on which see at Knizhnik-Zamolodchikov equation.

References

General

• Wikipedia arrangement of hyperplanes

• eom: arrangement of hyperplanes

• Peter Orlik, Hiroaki Terao, Arrangements of Hyperplanes, Grundlehren der Mathematischen Wissenschaften 300, Springer 1992, MR1217488

• I. M. Gelfand, M. M. Kapranov, A. Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhäuser 1994, 523 pp.

• Corrado De Concini, Claudio Procesi, Topics in hyperplane arrangements, polytopes and box-splines, Universitext 223, Springer 2010.

Stanley’s survey focuses instead more on combinatorics of the intersection poset of an arrangement (as well as arrangements in the case of vector space over a finite field):

• Richard P. Stanley, An introduction to hyperplane arrangements, in: Geometric combinatorics, 389–496, IAS/Park City Math. Ser., 13, Amer. Math. Soc. (2007) [pdf, errata]

See also:

• Richard Randell, Morse theory, Milnor fibers and minimality of hyperplane arrangements [math.AG/0011101]

• Daniel C. Cohen, Michael Falk, Richard Randell: Discriminantal bundles, arrangement groups, and subdirect products of free groups [arxiv/1008.0417]

• Takuya Saito, So Yamagata: Non-vanishing of homotopy groups of Manin–Schechtman arrangements [arXiv:2605.14536]

Relating to configuration spaces of points

See the references on Braid representatioons via twisted de Rham cohomology of configuration spaces