View source for Arrangement of hyperplanes

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107001.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107002.png" />-dimensional [[Affine space|affine space]] over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107003.png" />. An arrangement of hyperplanes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107004.png" />, is a finite collection of codimension-one affine subspaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107005.png" />, [[#References|[a5]]].

===Examples of arrangements of hyperplanes.===


1) A subset of the coordinate hyperplanes is called a Boolean arrangement.

2) An arrangement is in general position if at each point it is locally Boolean.

3) The braid arrangement consists of the hyperplanes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107006.png" />. It is the set of reflecting hyperplanes of the [[Symmetric group|symmetric group]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107007.png" /> letters.

4) The reflecting hyperplanes of a finite reflection group.

==Combinatorics.==
An edge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107008.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107009.png" /> is a non-empty intersection of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070010.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070011.png" /> be the set of edges partially ordered by reverse inclusion. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070012.png" /> is a geometric semi-lattice with minimal element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070013.png" />, rank given by codimension, and maximal elements of the same rank, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070014.png" />. The [[Möbius function|Möbius function]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070015.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070016.png" /> and, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070017.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070018.png" /></td> </tr></table>

The characteristic polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070019.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070020.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070021.png" />. For a generic arrangement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070022.png" /> hyperplanes,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070023.png" /></td> </tr></table>

For the braid arrangement,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070024.png" /></td> </tr></table>

Similar factorizations hold for all reflection arrangements involving the (co)exponents of the reflection group. Given a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070025.png" />-tuple of hyperplanes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070026.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070027.png" />; note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070028.png" /> may be empty. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070029.png" /> is dependent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070031.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070032.png" /> be the exterior algebra on symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070033.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070034.png" />, where the product is juxtaposition. Define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070035.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070037.png" /> and, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070038.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070039.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070040.png" /> be the [[Ideal|ideal]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070041.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070042.png" />. The Orlik–Solomon algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070043.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070044.png" />. For connections with matroid theory, see [[#References|[a3]]].

==Divisor.==
The divisor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070045.png" /> is the union of the hyperplanes, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070046.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070047.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070048.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070049.png" /> has the homotopy type of a wedge of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070050.png" /> spheres of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070051.png" />, [[#References|[a4]]]. The singularities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070052.png" /> are not isolated. The divisor of a general-position arrangement has normal crossings, but this is not true for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070053.png" />. Blowing-up <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070054.png" /> along all edges where it is not locally a product of arrangements yields a normal crossing divisor. See also [[Divisor|Divisor]].

==Complement.==
The complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070055.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070056.png" />.

1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070057.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070058.png" /> is a finite set of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070059.png" />.

2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070060.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070061.png" /> is a disjoint union of open convex sets (chambers) of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070062.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070064.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070065.png" /> chambers with compact closure, [[#References|[a7]]].

3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070066.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070067.png" /> is an open complex (Stein) manifold of the homotopy type of a finite [[CW-complex|CW-complex]] (cf. also [[Stein manifold|Stein manifold]]). Its [[Cohomology|cohomology]] is torsion-free and its Poincaré polynomial (cf. [[Künneth formula|Künneth formula]]) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070068.png" />. The product structure is determined by the isomorphism of graded algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070069.png" />. The [[Fundamental group|fundamental group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070070.png" /> has an effective presentation, but the higher homotopy groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070071.png" /> are not known in general.

The complement of a Boolean arrangement is a complex torus. In a general-position arrangement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070072.png" /> hyperplanes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070073.png" /> has non-trivial higher homotopy groups. For the braid arrangement, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070074.png" /> is called the pure braid space and its higher homotopy groups are trivial. The symmetric group acts freely on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070075.png" /> with as orbit space the braid space whose fundamental group is the braid group. The quotient of the divisor by the symmetric group is called the discriminant, which has connections with singularity theory.

==Ball quotients.==
Examples of algebraic surfaces whose universal cover is the complex ball were constructed as "Kummer" covers of the projective plane branched along certain arrangements of projective lines, [[#References|[a2]]].

==Logarithmic forms.==
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070076.png" />, choose a linear polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070077.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070078.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070079.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070080.png" /> denote all global regular (i.e., polynomial) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070081.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070082.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070083.png" /> denote the space of all global rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070084.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070085.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070086.png" /> of logarithmic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070087.png" />-forms with poles along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070088.png" /> is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070089.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070090.png" /></td> </tr></table>

The arrangement is free if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070091.png" /> is a free module over the polynomial ring. A free arrangement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070092.png" /> has integer exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070093.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070094.png" />. Reflection arrangements are free. This explains the factorization of their characteristic polynomials.

==Hypergeometric integrals.==
Certain rank-one local system cohomology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070095.png" /> may be identified with spaces of hypergeometric integrals, [[#References|[a1]]]. If the local system is suitably generic, these cohomology groups may be computed using the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070096.png" />. Only the top cohomology group is non-zero, and it has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070097.png" />. See [[#References|[a6]]] for connections with the representation theory of Lie algebras and [[Quantum groups|quantum groups]], and with the Knizhnik–Zamolodchikov differential equations of physics.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Aomoto, M. Kita, "Hypergeometric functions" , Springer (1994) (Translated from Japanese) {{MR|2799182}} {{MR|2482635}} {{MR|1866164}} {{MR|1805969}} {{MR|1792063}} {{MR|1768923}} {{MR|1803881}} {{MR|1749398}} {{MR|1614401}} {{MR|1401610}} {{MR|1256465}} {{MR|0988314}} {{ZBL|1229.33001}} {{ZBL|1169.33307}} {{ZBL|1174.33301}} {{ZBL|1189.33028}} {{ZBL|0972.33009}} {{ZBL|0927.33014}} {{ZBL|0943.32010}} {{ZBL|0787.33001}} {{ZBL|0859.33001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Barthel, F. Hirzebruch, T. Höfer, "Geradenkonfigurationen und Algebraische Flächen" , Vieweg (1987) {{MR|0912097}} {{ZBL|0645.14016}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Björner, M. Las Vergnas, B. Sturmfels, N. White, G.M. Ziegler, "Oriented matroids" , Cambridge Univ. Press (1993) {{MR|1226888}} {{ZBL|0773.52001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Goresky, R. MacPherson, "Stratified Morse theory" , Springer (1988) {{MR|0932724}} {{ZBL|0639.14012}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> P. Orlik, H. Terao, "Arrangements of hyperplanes" , Springer (1992) {{MR|1217488}} {{ZBL|0757.55001}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> A. Varchenko, "Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups" , World Sci. (1995) {{MR|1384760}} {{ZBL|0951.33001}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> T. Zaslavsky, "Facing up to arrangements: face-count formulas for partitions of space by hyperplanes" , ''Memoirs'' , '''154''' , Amer. Math. Soc. (1975) {{MR|0357135}} {{ZBL|0296.50010}} </TD></TR></table>