Ordered Hyperplane Arrangements¶

The HyperplaneArrangements orders the hyperplanes in a arrangement independently of the way the hyperplanes are introduced. The class OrderedHyperplaneArrangements fixes an order specified by the user. This can be needed for certain properties, e.g., fundamental group with information about meridians, braid monodromy with information about the strands; in the future, it may be useful for combinatorial properties. There are no other differences with usual hyperplane arrangements.

An ordered arrangement is an arrangement where the hyperplanes are sorted by the user:

sage: H0.<t0, t1, t2> = HyperplaneArrangements(QQ)
sage: H0(t0 - t1, t1 - t2, t0 - t2)
Arrangement <t1 - t2 | t0 - t1 | t0 - t2>
sage: H.<t0, t1, t2> = OrderedHyperplaneArrangements(QQ)
sage: H(t0 - t1, t1 - t2, t0 - t2)
Arrangement <t0 - t1 | t1 - t2 | t0 - t2>

Some methods are adapted, e.g., hyperplanes(), and some new ones are created, regarding hyperplane sections and fundamental groups:

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: H1.<x,y> = OrderedHyperplaneArrangements(QQ)
sage: A1 = H1(x, y); A = H(A1)
sage: A.hyperplanes()
(Hyperplane 0*x + y + 0, Hyperplane x + 0*y + 0)
sage: A1.hyperplanes()
(Hyperplane x + 0*y + 0, Hyperplane 0*x + y + 0)

We see the differences in union():

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: H1.<x,y> = OrderedHyperplaneArrangements(QQ)
sage: A = H([1,2,3], [0,1,1], [0,1,-1], [1,-1,0], [1,1,0])
sage: B = H([1,1,1], [1,-1,1], [1,0,-1])
sage: C = A.union(B)
sage: A1 = H1(A); B1 = H1(B); C1 = A1.union(B1)
sage: [C1.hyperplanes().index(h) for h in C.hyperplanes()]
[0, 5, 6, 1, 2, 3, 7, 4]

Also in meth: $s a g e . g e o m e t r y . h y p e r p l a n e_{a} r r a n g e m e n t . a r r a n g e m e n t . H y p e r p l a n e A r r a n g e m e n t E l e m e n t . c o n e$ :

sage: # needs sage.combinat
sage: a.<x,y,z> = hyperplane_arrangements.semiorder(3)
sage: H.<x,y,z> = OrderedHyperplaneArrangements(QQ)
sage: a1 = H(a)
sage: b = a.cone(); b1 = a1.cone()
sage: [b1.hyperplanes().index(h) for h in b.hyperplanes()]
[0, 2, 4, 6, 1, 3, 5]

And in restriction():

sage: # needs sage.graphs
sage: A.<u, x, y, z> = hyperplane_arrangements.braid(4)
sage: L.<u, x, y, z> = OrderedHyperplaneArrangements(QQ)
sage: A1 = L(A)
sage: H = A[0]; H
Hyperplane 0*u + 0*x + y - z + 0
sage: A.restriction(H)
Arrangement <x - z | u - x | u - z>
sage: A1.restriction(H)
Arrangement <x - z | u - x | u - z>
sage: A1.restriction(H, repetitions=True)
Arrangement of 5 hyperplanes of dimension 3 and rank 2

AUTHORS:

Enrique Artal (2023-12): initial version

This module adds some features to the unordered one for some properties which depend on the order.

class sage.geometry.hyperplane_arrangement.ordered_arrangement.OrderedHyperplaneArrangementElement(parent, hyperplanes, check=True, backend=None)[source]¶

Bases: HyperplaneArrangementElement

An ordered hyperplane arrangement.

Warning

You should never create OrderedHyperplaneArrangementElement instances directly, always use the parent.

affine_fundamental_group()[source]¶

Return the fundamental group of the complement of an affine hyperplane arrangement in $C^{n}$ whose equations have coefficients in a subfield of $\overset{―}{Q}$ .

OUTPUT: a finitely presented fundamental group

Note

This functionality requires the sirocco package to be installed.

EXAMPLES:

sage: # needs sirocco
sage: A.<x, y> = OrderedHyperplaneArrangements(QQ)
sage: L = [y + x, y + x - 1]
sage: H = A(L)
sage: H.affine_fundamental_group()
Finitely presented group < x0, x1 |  >
sage: L = [x, y, x + 1, y + 1, x - y]
sage: A(L).affine_fundamental_group()
Finitely presented group
< x0, x1, x2, x3, x4 | x4*x0*x4^-1*x0^-1,
                       x0*x2*x3*x2^-1*x0^-1*x3^-1,
                       x1*x2*x4*x2^-1*x1^-1*x4^-1,
                       x2*x3*x0*x2^-1*x0^-1*x3^-1,
                       x2*x4*x1*x2^-1*x1^-1*x4^-1,
                       x4*x1*x4^-1*x3^-1*x2^-1*x1^-1*x2*x3 >
sage: H = A(x, y, x + y)
sage: H.affine_fundamental_group()
Finitely presented group
< x0, x1, x2 | x0*x1*x2*x1^-1*x0^-1*x2^-1, x1*x2*x0*x1^-1*x0^-1*x2^-1 >
sage: H.affine_fundamental_group()  # repeat to use the attribute
Finitely presented group
< x0, x1, x2 | x0*x1*x2*x1^-1*x0^-1*x2^-1, x1*x2*x0*x1^-1*x0^-1*x2^-1 >
sage: T.<t> = QQ[]
sage: K.<a> = NumberField(t^3 + t + 1)
sage: L.<x, y> = OrderedHyperplaneArrangements(K)
sage: H = L(a*x + y -1, x + a*y + 1, x - 1, y - 1)
sage: H.affine_fundamental_group()
Traceback (most recent call last):
...
TypeError: the base field is not in QQbar
sage: L.<t> = OrderedHyperplaneArrangements(QQ)
sage: L([t - j for j in range(4)]).affine_fundamental_group()
Finitely presented group < x0, x1, x2, x3 |  >
sage: L.<x, y, z> = OrderedHyperplaneArrangements(QQ)
sage: L(L.gens() + (x + y + z + 1,)).affine_fundamental_group().sorted_presentation()
Finitely presented group
< x0, x1, x2, x3 | x3^-1*x2^-1*x3*x2, x3^-1*x1^-1*x3*x1,
                   x3^-1*x0^-1*x3*x0, x2^-1*x1^-1*x2*x1,
                   x2^-1*x0^-1*x2*x0, x1^-1*x0^-1*x1*x0 >
sage: A = OrderedHyperplaneArrangements(QQ, names=())
sage: H = A(); H
Empty hyperplane arrangement of dimension 0
sage: H.affine_fundamental_group()
Finitely presented group <  |  >

affine_meridians()[source]¶

Return the meridians of each hyperplane (including the one at infinity).

OUTPUT: a dictionary

Note

This functionality requires the sirocco package to be installed.

EXAMPLES:

sage: # needs sirocco
sage: A.<x, y> = OrderedHyperplaneArrangements(QQ)
sage: L = [y + x, y + x - 1]
sage: H = A(L)
sage: g = H.affine_fundamental_group()
sage: g
Finitely presented group < x0, x1 |  >
sage: H.affine_meridians()
{0: [x0], 1: [x1], 2: [x1^-1*x0^-1]}
sage: H1 = H.add_hyperplane(y - x)
sage: H1.affine_meridians()
{0: [x0], 1: [x1], 2: [x2], 3: [x2^-1*x1^-1*x0^-1]}

hyperplane_section(proj=True)[source]¶

Compute a generic hyperplane section of self.

INPUT:

proj – (default: True) if the ambient space is affine or projective

OUTPUT:

An arrangement $A$ obtained by intersecting with a generic hyperplane

EXAMPLES:

sage: L.<x, y, z> = OrderedHyperplaneArrangements(QQ)
sage: L(x, y - 1, z).hyperplane_section()
Traceback (most recent call last):
...
TypeError: the arrangement is not projective

sage: # needs sage.graphs
sage: A0.<u,x,y,z> = hyperplane_arrangements.braid(4); A0
Arrangement of 6 hyperplanes of dimension 4 and rank 3
sage: L.<u,x,y,z> = OrderedHyperplaneArrangements(QQ)
sage: A = L(A0)
sage: M = A.matroid()
sage: A1 = A.hyperplane_section()
sage: A1
Arrangement of 6 hyperplanes of dimension 3 and rank 3
sage: M1 = A1.matroid()
sage: A2 = A1.hyperplane_section(); A2
Arrangement of 6 hyperplanes of dimension 2 and rank 2
sage: M2 = A2.matroid()
sage: T1 = M1.truncation()
sage: T1.is_isomorphic(M2)
True
sage: T1.isomorphism(M2)
{0: 0, 1: 1, 2: 2, 3: 3, 4: 4, 5: 5}

sage: # needs sage.combinat
sage: a0 = hyperplane_arrangements.semiorder(3); a0
Arrangement of 6 hyperplanes of dimension 3 and rank 2
sage: L.<t0, t1, t2> = OrderedHyperplaneArrangements(QQ)
sage: a = L(a0)
sage: ca = a.cone()
sage: m = ca.matroid()
sage: a1 = a.hyperplane_section(proj=False)
sage: a1
Arrangement of 6 hyperplanes of dimension 2 and rank 2
sage: ca1 = a1.cone()
sage: m1 = ca1.matroid()
sage: m.isomorphism(m1)
{0: 0, 1: 1, 2: 2, 3: 3, 4: 4, 5: 5, 6: 6}
sage: p0 = hyperplane_arrangements.Shi(4)
sage: L.<t0, t1, t2, t3> = OrderedHyperplaneArrangements(QQ)
sage: p = L(p0)
sage: a = p.hyperplane_section(proj=False); a
Arrangement of 12 hyperplanes of dimension 3 and rank 3
sage: ca = a.cone()
sage: m = ca.matroid().truncation()
sage: a1 = a.hyperplane_section(proj=False); a1
Arrangement of 12 hyperplanes of dimension 2 and rank 2
sage: ca1 = a1.cone()
sage: m1 = ca1.matroid()
sage: m1.is_isomorphism(m, {j: j for j in range(13)})
True

projective_fundamental_group()[source]¶

Return the fundamental group of the complement of a projective hyperplane arrangement.

OUTPUT:

The finitely presented group of the complement in the projective space whose equations have coefficients in a subfield of $\overset{―}{Q}$ .

Note

This functionality requires the sirocco package to be installed.

EXAMPLES:

sage: # needs sirocco
sage: A.<x, y> = OrderedHyperplaneArrangements(QQ)
sage: H = A(x, y, x + y)
sage: H.projective_fundamental_group()
Finitely presented group < x0, x1 |  >

sage: # needs sirocco sage.graphs
sage: A3.<x, y, z> = OrderedHyperplaneArrangements(QQ)
sage: H = A3(hyperplane_arrangements.braid(4).essentialization())
sage: G3 = H.projective_fundamental_group(); G3.sorted_presentation()
Finitely presented group
< x0, x1, x2, x3, x4 | x4^-1*x3^-1*x2^-1*x3*x4*x0*x2*x0^-1,
                       x4^-1*x2^-1*x4*x2, x4^-1*x1^-1*x0^-1*x1*x4*x0,
                       x4^-1*x1^-1*x0^-1*x4*x0*x1,
                       x3^-1*x2^-1*x1^-1*x0^-1*x3*x0*x1*x2,
                       x3^-1*x1^-1*x3*x1 >
sage: G3.abelian_invariants()
(0, 0, 0, 0, 0)
sage: A4.<t1, t2, t3, t4> = OrderedHyperplaneArrangements(QQ)
sage: H = A4(hyperplane_arrangements.braid(4))
sage: G4 = H.projective_fundamental_group(); G4.sorted_presentation()  # long time (:issue:`39569`)
Finitely presented group
< x0, x1, x2, x3, x4 | x4^-1*x3^-1*x2^-1*x3*x4*x0*x2*x0^-1,
                       x4^-1*x2^-1*x4*x2, x4^-1*x1^-1*x0^-1*x1*x4*x0,
                       x4^-1*x1^-1*x0^-1*x4*x0*x1,
                       x3^-1*x2^-1*x1^-1*x0^-1*x3*x0*x1*x2, x3^-1*x1^-1*x3*x1 >
sage: G4.abelian_invariants()  # long time (:issue:`39569`)
(0, 0, 0, 0, 0)

sage: # needs sirocco
sage: L.<t0, t1, t2, t3, t4> = OrderedHyperplaneArrangements(QQ)
sage: H = hyperplane_arrangements.coordinate(5)
sage: H = L(H)
sage: g = H.projective_fundamental_group()
sage: g.is_abelian(), g.abelian_invariants()
(True, (0, 0, 0, 0))
sage: L(t0, t1, t2, t3, t4, t0 - 1).projective_fundamental_group()
Traceback (most recent call last):
...
TypeError: the arrangement is not projective
sage: T.<t> = QQ[]
sage: K.<a> = NumberField(t^3 + t + 1)
sage: L.<x, y, z> = OrderedHyperplaneArrangements(K)
sage: H = L(a*x + y - z, x + a*y + z, x - z, y - z)
sage: H.projective_fundamental_group()
Traceback (most recent call last):
...
TypeError: the base field is not in QQbar
sage: A.<x> = OrderedHyperplaneArrangements(QQ)
sage: H = A(); H
Empty hyperplane arrangement of dimension 1
sage: H.projective_fundamental_group()
Finitely presented group <  |  >

projective_meridians()[source]¶

Return the meridian of each hyperplane.

OUTPUT: a dictionary

Note

This functionality requires the sirocco package to be installed.

EXAMPLES:

sage: # needs sirocco
sage: A.<x, y> = OrderedHyperplaneArrangements(QQ)
sage: H = A(x, y, x + y)
sage: H.projective_meridians()
{0: x0, 1: x1, 2: [x1^-1*x0^-1]}

sage: # needs sirocco sage.graphs
sage: A3.<x, y, z> = OrderedHyperplaneArrangements(QQ)
sage: H = A3(hyperplane_arrangements.braid(4).essentialization())
sage: H.projective_meridians()
{0: [x2^-1*x0^-1*x4^-1*x3^-1*x1^-1],
 1: [x3], 2: [x4], 3: [x1], 4: [x2], 5: [x0]}
sage: A4.<t1, t2, t3, t4> = OrderedHyperplaneArrangements(QQ)
sage: H = A4(hyperplane_arrangements.braid(4))
sage: H.projective_meridians()  # long time
{0: [x2^-1*x0^-1*x4^-1*x3^-1*x1^-1], 1: [x3],
 2: [x4], 3: [x0], 4: [x2], 5: [x1]}

sage: # needs sirocco
sage: L.<t0, t1, t2, t3, t4> = OrderedHyperplaneArrangements(QQ)
sage: H = hyperplane_arrangements.coordinate(5)
sage: H = L(H)
sage: H.projective_meridians()
{0: [x2], 1: [x3], 2: [x0], 3: [x3^-1*x2^-1*x1^-1*x0^-1], 4: [x1]}

class sage.geometry.hyperplane_arrangement.ordered_arrangement.OrderedHyperplaneArrangements(base_ring, names=())[source]¶

Bases: HyperplaneArrangements

Ordered Hyperplane arrangements.

For more information on hyperplane arrangements, see sage.geometry.hyperplane_arrangement.arrangement.

INPUT:

base_ring – ring; the base ring
names – tuple of strings; the variable names

EXAMPLES:

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: x
Hyperplane x + 0*y + 0
sage: x + y
Hyperplane x + y + 0
sage: H(x, y, x-1, y-1)
Arrangement <y - 1 | y | x - 1 | x>

Element[source]¶: alias of OrderedHyperplaneArrangementElement