A hyperplane is a linear subspace of codimension one. An arrangement is a finite set of hyperplanes.
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 = A3
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 = A3'
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 = trivial
o7 : Hyperplane Arrangement
|
i8 : describe trivial
o8 = {}
|
i9 : ring trivial
o9 = S
o9 : PolynomialRing
|