next | previous | forward | backward | up | top | index | toc | Macaulay2 website
HyperplaneArrangements :: arrangement

arrangement -- create a hyperplane arrangement

Synopsis

Description

A hyperplane is an affine-linear subspace of codimension one. An arrangement is a finite set of hyperplanes. If each hyperplane contains the origin, the arrangement is a central arrangement.

Probably the best-known hyperplane arrangement is the braid arrangement consisting of all the diagonal hyperplanes. In 4-space, it is constructed as follows:

i1 : S = ZZ[w,x,y,z];
i2 : A3 = arrangement {w-x,w-y,w-z,x-y,x-z,y-z}

o2 = {w - x, w - y, w - z, x - y, x - z, y - z}

o2 : Hyperplane Arrangement 
i3 : describe A3

o3 = {w - x, w - y, w - z, x - y, x - z, y - z}
If we project along onto a subspace, then we obtain an essential arrangement:
i4 : R = S/ideal(w+x+y+z)

o4 = R

o4 : QuotientRing
i5 : A3' = arrangement({w-x,w-y,w-z,x-y,x-z,y-z},R)

o5 = {- 2x - y - z, - x - 2y - z, - x - y - 2z, x - y, x - z, y - z}

o5 : Hyperplane Arrangement 
i6 : describe A3'

o6 = {- 2x - y - z, - x - 2y - z, - x - y - 2z, x - y, x - z, y - z}
The trivial arrangement has no equations.
i7 : trivial = arrangement({},S)

o7 = {}

o7 : Hyperplane Arrangement 
i8 : describe trivial

o8 = {}
i9 : ring trivial

o9 = S

o9 : PolynomialRing
i10 : use S;
i11 : arrangement (x^2*y^2*(x^2-y^2)*(x^2-z^2))

o11 = {y, y, x, x, x - z, x + z, x - y, x + y}

o11 : Hyperplane Arrangement 

Caveat

If the elements of L are not ring elements in R, then the induced identity map is used to map them from ring L#0 into R.

If arrangement Q is used, the order of the factors is determined internally.

See also

Ways to use arrangement :

For the programmer

The object arrangement is a method function.