Affine and Projective Plane Curve Arrangements#

We create classes AffinePlaneCurveArrangements and ProjectivePlaneCurveArrangements following the properties of HyperplaneArrangements:

sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: C = H(3*x + 2*y - x^2 + y^3 - 7);  C
Arrangement (y^3 - x^2 + 3*x + 2*y - 7) in Affine Space of dimension 2 over Rational Field

>>> from sage.all import *
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> C = H(Integer(3)*x + Integer(2)*y - x**Integer(2) + y**Integer(3) - Integer(7));  C
Arrangement (y^3 - x^2 + 3*x + 2*y - 7) in Affine Space of dimension 2 over Rational Field

The individual curves will be in AffinePlaneCurve or in ProjectivePlaneCurve:

sage: C[0].parent()
<class 'sage.schemes.curves.affine_curve.IntegralAffinePlaneCurve_with_category'>

>>> from sage.all import *
>>> C[Integer(0)].parent()
<class 'sage.schemes.curves.affine_curve.IntegralAffinePlaneCurve_with_category'>

The default base field is \(\QQ\), the rational numbers. Number fields are also possible (also with fixed embeddings in \(\QQbar\)):

sage: # needs sage.rings.number_field
sage: x = polygen(QQ, 'x')
sage: NF.<a> = NumberField(x^4 - 5 * x^2 + 5, embedding=1.90)
sage: H.<y,z> = AffinePlaneCurveArrangements(NF)
sage: A = H(y^2 - a * z, y^2 + a * z); A
Arrangement (y^2 + (-a)*z, y^2 + a*z) in Affine Space of dimension 2
over Number Field in a with defining polynomial
x^4 - 5*x^2 + 5 with a = 1.902113032590308?
sage: A.base_ring()
Number Field in a with defining polynomial x^4 - 5*x^2 + 5
with a = 1.902113032590308?

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(QQ, 'x')
>>> NF = NumberField(x**Integer(4) - Integer(5) * x**Integer(2) + Integer(5), embedding=RealNumber('1.90'), names=('a',)); (a,) = NF._first_ngens(1)
>>> H = AffinePlaneCurveArrangements(NF, names=('y', 'z',)); (y, z,) = H._first_ngens(2)
>>> A = H(y**Integer(2) - a * z, y**Integer(2) + a * z); A
Arrangement (y^2 + (-a)*z, y^2 + a*z) in Affine Space of dimension 2
over Number Field in a with defining polynomial
x^4 - 5*x^2 + 5 with a = 1.902113032590308?
>>> A.base_ring()
Number Field in a with defining polynomial x^4 - 5*x^2 + 5
with a = 1.902113032590308?

AUTHORS:

  • Enrique Artal (2023-10): initial version

class sage.schemes.curves.plane_curve_arrangement.AffinePlaneCurveArrangementElement(parent, curves, check=True)[source]#

Bases: PlaneCurveArrangementElement

An ordered affine plane curve arrangement.

braid_monodromy(vertical=True)[source]#

Return the braid monodromy of the complement of the union of affine plane curves in \(\CC^2\). If there are vertical asymptotes a change of variable is done.

INPUT:

  • vertical – boolean (default: True); if it is True, there are no vertical asymptotes, and there are vertical lines, then a simplified braid_monodromy() is computed.

OUTPUT:

A braid monodromy with dictionnaries identifying strands with components and braids with vertical lines.

Note

This functionality requires the sirocco package to be installed.

EXAMPLES:

sage: # needs sirocco
sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: A = H(y^2 + x, y + x - 1, x)
sage: A.braid_monodromy(vertical=False)
[s1*s0*(s1*s2*s1)^2*s2*(s1^-1*s2^-1)^2*s1^-1*s0^-1*s1^-1,
 s1*s0*(s1*s2)^2*s2*s1^-1*s2^-1*s1^-1*s0^-1*s1^-1,
 s1*s0*s1*s2*(s1*s2^-1)^2*s0*s1*s2*s1*s0*s2^-1*s1^-3*s2*s1^-1*s2^-1*s1^-1*s0^-1*s1^-1,
 s1*s0*s1*s2*s1*s2^-1*s1^4*s2*s1^-1*s2^-1*s1^-1*s0^-1*s1^-1,
 s1*s0*s1*s2*s1*s2^-1*s1^-1*s2*s0^-1*s1^-1]
sage: A.braid_monodromy(vertical=True)
[s1*s0*s1*s0^-1*s1^-1*s0, s0^-1*s1*s0*s1^-1*s0, s0^-1*s1^2*s0]

>>> from sage.all import *
>>> # needs sirocco
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H(y**Integer(2) + x, y + x - Integer(1), x)
>>> A.braid_monodromy(vertical=False)
[s1*s0*(s1*s2*s1)^2*s2*(s1^-1*s2^-1)^2*s1^-1*s0^-1*s1^-1,
 s1*s0*(s1*s2)^2*s2*s1^-1*s2^-1*s1^-1*s0^-1*s1^-1,
 s1*s0*s1*s2*(s1*s2^-1)^2*s0*s1*s2*s1*s0*s2^-1*s1^-3*s2*s1^-1*s2^-1*s1^-1*s0^-1*s1^-1,
 s1*s0*s1*s2*s1*s2^-1*s1^4*s2*s1^-1*s2^-1*s1^-1*s0^-1*s1^-1,
 s1*s0*s1*s2*s1*s2^-1*s1^-1*s2*s0^-1*s1^-1]
>>> A.braid_monodromy(vertical=True)
[s1*s0*s1*s0^-1*s1^-1*s0, s0^-1*s1*s0*s1^-1*s0, s0^-1*s1^2*s0]
fundamental_group(simplified=True, vertical=True, projective=False)[source]#

Return the fundamental group of the complement of the union of affine plane curves in \(\CC^2\).

INPUT:

  • vertical – boolean (default: True); if True, there are no vertical asymptotes, and there are vertical lines, then a simplified braid braid_monodromy() is used

  • simplified – boolean (default: True); if it is True, the group is simplified

  • projective – boolean (default: False); to be used in the method for projective curves

OUTPUT:

A finitely presented group.

Note

This functionality requires the sirocco package to be installed.

EXAMPLES:

sage: # needs sirocco
sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: A = H(y^2 + x, y + x - 1, x)
sage: A.fundamental_group()
Finitely presented group
< x0, x1, x2 | x2*x0*x2^-1*x0^-1, x1*x0*x1^-1*x0^-1, (x2*x1)^2*(x2^-1*x1^-1)^2 >
sage: A.meridians()
{0: [x1, x2*x1*x2^-1], 1: [x0], 2: [x2],
 3: [x1^-1*x2^-1*x1^-1*x0^-1]}
sage: G = A.fundamental_group(simplified=False)
sage: G.sorted_presentation()
Finitely presented group
< x0, x1, x2, x3 | x3^-1*x2^-1*x3*x0*x1*x0^-1,
                   x3^-1*x1^-1*x3*x0*x1*x0^-1*x2^-1*x0^-1*(x2*x0)^2*x1^-1*x0^-1,
                   x3^-1*x0^-1*x3*x0*x1*x0^-1*x2^-1*x0*x2*x0*x1^-1*x0^-1,
                   x2^-1*x0^-1*x2*x0, x1^-1*x0^-1*x1*x0 >
sage: A.meridians(simplified=False)
{0: [x1, x2], 1: [x0], 2: [x3], 3: [x3^-1*x2^-1*x1^-1*x0^-1]}
sage: A.fundamental_group(vertical=False)
Finitely presented group
< x0, x1, x2 | x2^-1*x1^-1*x2*x1, x1*x0*x1^-1*x0^-1, (x0*x2)^2*(x0^-1*x2^-1)^2 >
sage: A.meridians(vertical=False)
{0: [x2, x0*x2*x0^-1], 1: [x1], 2: [x0], 3: [x0*x2^-1*x0^-1*x2^-1*x1^-1*x0^-1]}
sage: G = A.fundamental_group(simplified=False, vertical=False)
sage: G.sorted_presentation()
Finitely presented group
< x0, x1, x2, x3 | x3^-1*x2^-1*x1^-1*x2*x3*x2^-1*x1*x2,
                   x3^-1*x2^-1*x1^-1*x2*x3*x2^-1*x1*x2,
                   (x3^-1*x2^-1*x0^-1*x2)^2*(x3*x2^-1*x0*x2)^2,
                   x3^-1*x2^-1*x0^-1*x2*x3^-1*x2^-1*x0*x2*x3*x2,
                   x1^-1*x0^-1*x1*x0 >
sage: A.meridians(simplified=False, vertical=False)
{0: [x2, x3], 1: [x1], 2: [x0], 3: [x3^-1*x2^-1*x1^-1*x0^-1]}
sage: A = H(x * y^2 + x + y, y + x -1, x, y)
sage: G = A.fundamental_group()
sage: G.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 >

>>> from sage.all import *
>>> # needs sirocco
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H(y**Integer(2) + x, y + x - Integer(1), x)
>>> A.fundamental_group()
Finitely presented group
< x0, x1, x2 | x2*x0*x2^-1*x0^-1, x1*x0*x1^-1*x0^-1, (x2*x1)^2*(x2^-1*x1^-1)^2 >
>>> A.meridians()
{0: [x1, x2*x1*x2^-1], 1: [x0], 2: [x2],
 3: [x1^-1*x2^-1*x1^-1*x0^-1]}
>>> G = A.fundamental_group(simplified=False)
>>> G.sorted_presentation()
Finitely presented group
< x0, x1, x2, x3 | x3^-1*x2^-1*x3*x0*x1*x0^-1,
                   x3^-1*x1^-1*x3*x0*x1*x0^-1*x2^-1*x0^-1*(x2*x0)^2*x1^-1*x0^-1,
                   x3^-1*x0^-1*x3*x0*x1*x0^-1*x2^-1*x0*x2*x0*x1^-1*x0^-1,
                   x2^-1*x0^-1*x2*x0, x1^-1*x0^-1*x1*x0 >
>>> A.meridians(simplified=False)
{0: [x1, x2], 1: [x0], 2: [x3], 3: [x3^-1*x2^-1*x1^-1*x0^-1]}
>>> A.fundamental_group(vertical=False)
Finitely presented group
< x0, x1, x2 | x2^-1*x1^-1*x2*x1, x1*x0*x1^-1*x0^-1, (x0*x2)^2*(x0^-1*x2^-1)^2 >
>>> A.meridians(vertical=False)
{0: [x2, x0*x2*x0^-1], 1: [x1], 2: [x0], 3: [x0*x2^-1*x0^-1*x2^-1*x1^-1*x0^-1]}
>>> G = A.fundamental_group(simplified=False, vertical=False)
>>> G.sorted_presentation()
Finitely presented group
< x0, x1, x2, x3 | x3^-1*x2^-1*x1^-1*x2*x3*x2^-1*x1*x2,
                   x3^-1*x2^-1*x1^-1*x2*x3*x2^-1*x1*x2,
                   (x3^-1*x2^-1*x0^-1*x2)^2*(x3*x2^-1*x0*x2)^2,
                   x3^-1*x2^-1*x0^-1*x2*x3^-1*x2^-1*x0*x2*x3*x2,
                   x1^-1*x0^-1*x1*x0 >
>>> A.meridians(simplified=False, vertical=False)
{0: [x2, x3], 1: [x1], 2: [x0], 3: [x3^-1*x2^-1*x1^-1*x0^-1]}
>>> A = H(x * y**Integer(2) + x + y, y + x -Integer(1), x, y)
>>> G = A.fundamental_group()
>>> G.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 >
meridians(simplified=True, vertical=True)[source]#

Return the meridians of each irreducible component.

OUTPUT:

A dictionary which associates the index of each curve with its meridians, including the line at infinity if it can be omputed

Note

This functionality requires the sirocco package to be installed and AffinePlaneCurveArrangements.fundamental_group() with the same options, where some examples are shown.

EXAMPLES:

sage: # needs sirocco
sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: A = H(x-1, y, x, y - 1)
sage: A.fundamental_group()
Finitely presented group
< x0, x1, x2, x3 | x2*x0*x2^-1*x0^-1, x2*x1*x2^-1*x1^-1,
                   x3*x0*x3^-1*x0^-1, x3*x1*x3^-1*x1^-1 >
sage: A.meridians()
{0: [x2], 1: [x0], 2: [x3], 3: [x1], 4: [x3^-1*x2^-1*x1^-1*x0^-1]}

>>> from sage.all import *
>>> # needs sirocco
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H(x-Integer(1), y, x, y - Integer(1))
>>> A.fundamental_group()
Finitely presented group
< x0, x1, x2, x3 | x2*x0*x2^-1*x0^-1, x2*x1*x2^-1*x1^-1,
                   x3*x0*x3^-1*x0^-1, x3*x1*x3^-1*x1^-1 >
>>> A.meridians()
{0: [x2], 1: [x0], 2: [x3], 3: [x1], 4: [x3^-1*x2^-1*x1^-1*x0^-1]}
strands()[source]#

Return the strands for each member of the arrangement.

OUTPUT:

A dictionary which associates to the index of each strand its associated component if the braid monodromy has been calculated with vertical=False.

Note

This functionality requires the sirocco package to be installed.

EXAMPLES:

sage: # needs sirocco
sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: A = H(y^2 + x, y + x - 1, x)
sage: bm = A.braid_monodromy()
sage: A.strands()
{0: 2, 1: 1, 2: 0, 3: 0}

>>> from sage.all import *
>>> # needs sirocco
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H(y**Integer(2) + x, y + x - Integer(1), x)
>>> bm = A.braid_monodromy()
>>> A.strands()
{0: 2, 1: 1, 2: 0, 3: 0}
vertical_lines_in_braid_monodromy()[source]#

Return the vertical lines in the arrangement.

OUTPUT:

A dictionary which associates the index of a braid to the index of the vertical line associated to the braid.

Note

This functionality requires the sirocco package to be installed.

EXAMPLES:

sage: # needs sirocco
sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: A = H(y^2 + x, y + x - 1, x)
sage: A.vertical_lines_in_braid_monodromy()
{1: 2}
sage: A.braid_monodromy(vertical=True)
[s1*s0*s1*s0^-1*s1^-1*s0, s0^-1*s1*s0*s1^-1*s0, s0^-1*s1^2*s0]

>>> from sage.all import *
>>> # needs sirocco
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H(y**Integer(2) + x, y + x - Integer(1), x)
>>> A.vertical_lines_in_braid_monodromy()
{1: 2}
>>> A.braid_monodromy(vertical=True)
[s1*s0*s1*s0^-1*s1^-1*s0, s0^-1*s1*s0*s1^-1*s0, s0^-1*s1^2*s0]
vertical_strands()[source]#

Return the strands if the braid monodromy has been computed with the vertical option.

OUTPUT:

A dictionary which associates to the index of each strand its associated component if the braid monodromy has been calculated with vertical=True.

Note

This functionality requires the sirocco package to be installed.

EXAMPLES:

sage: # needs sirocco
sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: A = H(y^2 + x, y + x - 1, x)
sage: A.vertical_strands()
{0: 1, 1: 0, 2: 0}
sage: A.braid_monodromy(vertical=True)
[s1*s0*s1*s0^-1*s1^-1*s0, s0^-1*s1*s0*s1^-1*s0, s0^-1*s1^2*s0]

>>> from sage.all import *
>>> # needs sirocco
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H(y**Integer(2) + x, y + x - Integer(1), x)
>>> A.vertical_strands()
{0: 1, 1: 0, 2: 0}
>>> A.braid_monodromy(vertical=True)
[s1*s0*s1*s0^-1*s1^-1*s0, s0^-1*s1*s0*s1^-1*s0, s0^-1*s1^2*s0]
class sage.schemes.curves.plane_curve_arrangement.AffinePlaneCurveArrangements(base_ring, names=())[source]#

Bases: PlaneCurveArrangements

Affine curve arrangements.

INPUT:

  • base_ring – ring; the base ring

  • names – tuple of strings; the variable names

EXAMPLES:

sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: H(x, y^2, x-1, y-1)
Arrangement (x, y^2, x - 1, y - 1) in Affine Space
of dimension 2 over Rational Field

>>> from sage.all import *
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> H(x, y**Integer(2), x-Integer(1), y-Integer(1))
Arrangement (x, y^2, x - 1, y - 1) in Affine Space
of dimension 2 over Rational Field
Element[source]#

alias of AffinePlaneCurveArrangementElement

ambient_space()[source]#

Return the ambient space.

EXAMPLES:

sage: L.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: L.ambient_space()
Affine Space of dimension 2 over Rational Field

>>> from sage.all import *
>>> L = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> L.ambient_space()
Affine Space of dimension 2 over Rational Field
class sage.schemes.curves.plane_curve_arrangement.PlaneCurveArrangementElement(parent, curves, check=True)[source]#

Bases: Element

An ordered plane curve arrangement.

add_curves(other)[source]#

The union of self with other.

INPUT:

  • other – a curve arrangement or something that can be converted into a curve arrangement

OUTPUT:

A new curve arrangement.

EXAMPLES:

sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: h = H([x * y, x + y + 1, x^3 - y^5, x^2 * y^2 + x^5 + y^5, (x^2 + y^2)^3 + (x^3 + y^3 - 1)^2])
sage: C = Curve(x^8 - y^8 -x^4 * y^4)
sage: h1 = h.union(C); h1
Arrangement of 6 curves in Affine Space of dimension 2 over Rational Field
sage: h1 == h1.union(C)
True

>>> from sage.all import *
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> h = H([x * y, x + y + Integer(1), x**Integer(3) - y**Integer(5), x**Integer(2) * y**Integer(2) + x**Integer(5) + y**Integer(5), (x**Integer(2) + y**Integer(2))**Integer(3) + (x**Integer(3) + y**Integer(3) - Integer(1))**Integer(2)])
>>> C = Curve(x**Integer(8) - y**Integer(8) -x**Integer(4) * y**Integer(4))
>>> h1 = h.union(C); h1
Arrangement of 6 curves in Affine Space of dimension 2 over Rational Field
>>> h1 == h1.union(C)
True
change_ring(base_ring)[source]#

Return curve arrangement over the new base ring.

INPUT:

  • base_ring – the new base ring; must be a field for curve arrangements

OUTPUT:

The curve arrangement obtained by changing the base field, as a new curve arrangement.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: A = H(y^2 - x^3, x, y, y^2 + x * y + x^2)
sage: K.<a> = CyclotomicField(3)
sage: A.change_ring(K)
Arrangement (-x^3 + y^2, x, y, x^2 + x*y + y^2) in Affine Space of
dimension 2 over Cyclotomic Field of order 3 and degree 2

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H(y**Integer(2) - x**Integer(3), x, y, y**Integer(2) + x * y + x**Integer(2))
>>> K = CyclotomicField(Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> A.change_ring(K)
Arrangement (-x^3 + y^2, x, y, x^2 + x*y + y^2) in Affine Space of
dimension 2 over Cyclotomic Field of order 3 and degree 2
coordinate_ring()[source]#

Return the coordinate ring of self.

OUTPUT:

The coordinate ring of the curve arrangement.

EXAMPLES:

sage: L.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: C = L(x, y)
sage: C.coordinate_ring()
Multivariate Polynomial Ring in x, y over Rational Field
sage: P.<x, y, z> = ProjectivePlaneCurveArrangements(QQ)
sage: C = P(x, y)
sage: C.coordinate_ring()
Multivariate Polynomial Ring in x, y, z over Rational Field

>>> from sage.all import *
>>> L = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> C = L(x, y)
>>> C.coordinate_ring()
Multivariate Polynomial Ring in x, y over Rational Field
>>> P = ProjectivePlaneCurveArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> C = P(x, y)
>>> C.coordinate_ring()
Multivariate Polynomial Ring in x, y, z over Rational Field
curves()[source]#

Return the curves in the arrangement as a tuple.

OUTPUT:

A tuple.

EXAMPLES:

sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: h = H((x * y, x + y + 1))
sage: h.curves()
(Affine Plane Curve over Rational Field defined by x*y,
 Affine Plane Curve over Rational Field defined by x + y + 1)

>>> from sage.all import *
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> h = H((x * y, x + y + Integer(1)))
>>> h.curves()
(Affine Plane Curve over Rational Field defined by x*y,
 Affine Plane Curve over Rational Field defined by x + y + 1)

Note that the curves can be indexed as if they were a list:

sage: h[1]
Affine Plane Curve over Rational Field defined by x + y + 1

>>> from sage.all import *
>>> h[Integer(1)]
Affine Plane Curve over Rational Field defined by x + y + 1
defining_polynomial(simplified=True)[source]#

Return the defining polynomial of the union of the curves in self.

EXAMPLES:

sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: A = H(y ** 2 + x ** 2, x, y)
sage: prod(A.defining_polynomials()) == A.defining_polynomial()
True

>>> from sage.all import *
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H(y ** Integer(2) + x ** Integer(2), x, y)
>>> prod(A.defining_polynomials()) == A.defining_polynomial()
True
defining_polynomials()[source]#

Return the defining polynomials of the elements of self.

EXAMPLES:

sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: A = H(y^2 - x^3, x, y, y^2 + x * y + x^2)
sage: A.defining_polynomials()
(-x^3 + y^2, x, y, x^2 + x*y + y^2)

>>> from sage.all import *
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H(y**Integer(2) - x**Integer(3), x, y, y**Integer(2) + x * y + x**Integer(2))
>>> A.defining_polynomials()
(-x^3 + y^2, x, y, x^2 + x*y + y^2)
deletion(curves)[source]#

Return the curve arrangement obtained by removing curves.

INPUT:

  • curves – a curve or curve arrangement

OUTPUT:

A new curve arrangement with the given curve(s) h removed.

EXAMPLES:

sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: h = H([x * y, x + y + 1, x^3 - y^5, x^2 * y^2 + x^5 + y^5, (x^2 + y^2)^3 + (x^3 + y^3 - 1)^2])
sage: C = h[-1]
sage: h.deletion(C)
Arrangement (x*y, x + y + 1, -y^5 + x^3, x^5 + y^5 + x^2*y^2)
in Affine Space of dimension 2 over Rational Field
sage: h.deletion(x)
Traceback (most recent call last):
...
ValueError: curve is not in the arrangement

>>> from sage.all import *
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> h = H([x * y, x + y + Integer(1), x**Integer(3) - y**Integer(5), x**Integer(2) * y**Integer(2) + x**Integer(5) + y**Integer(5), (x**Integer(2) + y**Integer(2))**Integer(3) + (x**Integer(3) + y**Integer(3) - Integer(1))**Integer(2)])
>>> C = h[-Integer(1)]
>>> h.deletion(C)
Arrangement (x*y, x + y + 1, -y^5 + x^3, x^5 + y^5 + x^2*y^2)
in Affine Space of dimension 2 over Rational Field
>>> h.deletion(x)
Traceback (most recent call last):
...
ValueError: curve is not in the arrangement
have_common_factors()[source]#

Check if the curves have common factors.

EXAMPLES:

sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: A = H(x * y, x^2 + x* y^3)
sage: A.have_common_factors()
True
sage: H(x * y, x + y^3).have_common_factors()
False

>>> from sage.all import *
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H(x * y, x**Integer(2) + x* y**Integer(3))
>>> A.have_common_factors()
True
>>> H(x * y, x + y**Integer(3)).have_common_factors()
False
ncurves()[source]#

Return the number of curves in the arrangement.

OUTPUT:

An integer.

EXAMPLES:

sage: H.<x, y, z> = ProjectivePlaneCurveArrangements(QQ)
sage: h = H((x * y, x + y + z))
sage: h.ncurves()
2
sage: len(h)    # equivalent
2

>>> from sage.all import *
>>> H = ProjectivePlaneCurveArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = H._first_ngens(3)
>>> h = H((x * y, x + y + z))
>>> h.ncurves()
2
>>> len(h)    # equivalent
2
reduce(clean=False, verbose=False)[source]#

Replace the curves by their reduction.

INPUT:

  • clean – boolean (default: False); if False and there are common factors it returns None and a warning message. If True, the common factors are kept only in the first occurance.

EXAMPLES:

sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: A = H(y^2, (x + y)^3 * (x^2 + x * y + y^2))
sage: A.reduce()
Arrangement (y, x^3 + 2*x^2*y + 2*x*y^2 + y^3) in Affine Space
of dimension 2 over Rational Field
sage: C = H(x*y, x*(y + 1))
sage: C.reduce(verbose=True)
Some curves have common components
sage: C.reduce(clean=True)
Arrangement (x*y, y + 1) in Affine Space of dimension 2
over Rational Field
sage: C = H(x*y, x)
sage: C.reduce(clean=True)
Arrangement (x*y) in Affine Space of dimension 2 over Rational Field

>>> from sage.all import *
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H(y**Integer(2), (x + y)**Integer(3) * (x**Integer(2) + x * y + y**Integer(2)))
>>> A.reduce()
Arrangement (y, x^3 + 2*x^2*y + 2*x*y^2 + y^3) in Affine Space
of dimension 2 over Rational Field
>>> C = H(x*y, x*(y + Integer(1)))
>>> C.reduce(verbose=True)
Some curves have common components
>>> C.reduce(clean=True)
Arrangement (x*y, y + 1) in Affine Space of dimension 2
over Rational Field
>>> C = H(x*y, x)
>>> C.reduce(clean=True)
Arrangement (x*y) in Affine Space of dimension 2 over Rational Field
union(other)[source]#

The union of self with other.

INPUT:

  • other – a curve arrangement or something that can be converted into a curve arrangement

OUTPUT:

A new curve arrangement.

EXAMPLES:

sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: h = H([x * y, x + y + 1, x^3 - y^5, x^2 * y^2 + x^5 + y^5, (x^2 + y^2)^3 + (x^3 + y^3 - 1)^2])
sage: C = Curve(x^8 - y^8 -x^4 * y^4)
sage: h1 = h.union(C); h1
Arrangement of 6 curves in Affine Space of dimension 2 over Rational Field
sage: h1 == h1.union(C)
True

>>> from sage.all import *
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> h = H([x * y, x + y + Integer(1), x**Integer(3) - y**Integer(5), x**Integer(2) * y**Integer(2) + x**Integer(5) + y**Integer(5), (x**Integer(2) + y**Integer(2))**Integer(3) + (x**Integer(3) + y**Integer(3) - Integer(1))**Integer(2)])
>>> C = Curve(x**Integer(8) - y**Integer(8) -x**Integer(4) * y**Integer(4))
>>> h1 = h.union(C); h1
Arrangement of 6 curves in Affine Space of dimension 2 over Rational Field
>>> h1 == h1.union(C)
True
class sage.schemes.curves.plane_curve_arrangement.PlaneCurveArrangements(base_ring, names=())[source]#

Bases: UniqueRepresentation, Parent

Plane curve arrangements.

INPUT:

  • base_ring – ring; the base ring

  • names – tuple of strings; the variable names

EXAMPLES:

sage: H.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: H(x, y^2, x-1, y-1)
Arrangement (x, y^2, x - 1, y - 1) in Affine Space
of dimension 2 over Rational Field

>>> from sage.all import *
>>> H = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> H(x, y**Integer(2), x-Integer(1), y-Integer(1))
Arrangement (x, y^2, x - 1, y - 1) in Affine Space
of dimension 2 over Rational Field
Element[source]#

alias of PlaneCurveArrangementElement

ambient_space()[source]#

Return the ambient space.

EXAMPLES:

sage: L.<x, y> = PlaneCurveArrangements(QQ)
Traceback (most recent call last):
...
NotImplementedError: <abstract method ambient_space at  0x...>
sage: L.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: L.ambient_space()
Affine Space of dimension 2 over Rational Field
sage: L.<x, y, z> = ProjectivePlaneCurveArrangements(QQ)
sage: L.ambient_space()
Projective Space of dimension 2 over Rational Field

>>> from sage.all import *
>>> L = PlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
Traceback (most recent call last):
...
NotImplementedError: <abstract method ambient_space at  0x...>
>>> L = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> L.ambient_space()
Affine Space of dimension 2 over Rational Field
>>> L = ProjectivePlaneCurveArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = L._first_ngens(3)
>>> L.ambient_space()
Projective Space of dimension 2 over Rational Field
change_ring(base_ring)[source]#

Return curve arrangements over a different base ring.

INPUT:

  • base_ring – a ring; the new base ring.

OUTPUT:

A new PlaneCurveArrangements instance over the new base ring

EXAMPLES:

sage: L.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: L.change_ring(RR).base_ring()
Real Field with 53 bits of precision

>>> from sage.all import *
>>> L = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> L.change_ring(RR).base_ring()
Real Field with 53 bits of precision
coordinate_ring()[source]#

Return the coordinate ring.

OUTPUT:

The coordinate ring of the curve arrangement.

EXAMPLES:

sage: L.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: L.coordinate_ring()
Multivariate Polynomial Ring in x, y over Rational Field

>>> from sage.all import *
>>> L = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> L.coordinate_ring()
Multivariate Polynomial Ring in x, y over Rational Field
gen(i)[source]#

Return the \(i\)-th coordinate.

INPUT:

  • i – integer

OUTPUT:

A variable.

EXAMPLES:

sage: L.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: L.gen(1)
y
sage: L.<x, y, z> = ProjectivePlaneCurveArrangements(QQ)
sage: L.gen(2)
z

>>> from sage.all import *
>>> L = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> L.gen(Integer(1))
y
>>> L = ProjectivePlaneCurveArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = L._first_ngens(3)
>>> L.gen(Integer(2))
z
gens()[source]#

Return the coordinates.

OUTPUT:

A tuple of linear expressions, one for each linear variable.

EXAMPLES:

sage: L = AffinePlaneCurveArrangements(QQ, ('x', 'y'))
sage: L.gens()
(x, y)
sage: L = ProjectivePlaneCurveArrangements(QQ, ('x', 'y', 'z'))
sage: L.gens()
(x, y, z)

>>> from sage.all import *
>>> L = AffinePlaneCurveArrangements(QQ, ('x', 'y'))
>>> L.gens()
(x, y)
>>> L = ProjectivePlaneCurveArrangements(QQ, ('x', 'y', 'z'))
>>> L.gens()
(x, y, z)
ngens()[source]#

Return the number of variables, i.e. 2 or 3, kept for completness.

OUTPUT:

An integer, 2 or 3, depending if the arrangement is projective or affine.

EXAMPLES:

sage: L.<x, y> = AffinePlaneCurveArrangements(QQ)
sage: L.ngens()
2
sage: L.<x, y, z> = ProjectivePlaneCurveArrangements(QQ)
sage: L.ngens()
3

>>> from sage.all import *
>>> L = AffinePlaneCurveArrangements(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> L.ngens()
2
>>> L = ProjectivePlaneCurveArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = L._first_ngens(3)
>>> L.ngens()
3
class sage.schemes.curves.plane_curve_arrangement.ProjectivePlaneCurveArrangementElement(parent, curves, check=True)[source]#

Bases: PlaneCurveArrangementElement

An ordered projective plane curve arrangement.

fundamental_group(simplified=True)[source]#

Return the fundamental group of the complement of the union of an arragnement of projective plane curves in the projective plane.

INPUT:

  • simplified – boolean (default: True); set if the group is simplified

OUTPUT:

A finitely presented group.

Note

This functionality requires the sirocco package to be installed.

EXAMPLES:

sage: # needs sirocco
sage: H.<x, y, z> = ProjectivePlaneCurveArrangements(QQ)
sage: H(z).fundamental_group()
Finitely presented group <  |  >
sage: H(x*y).fundamental_group()
Finitely presented group < x |  >
sage: A = H(y^2 + x*z, y + x - z, x)
sage: A.fundamental_group().sorted_presentation()
Finitely presented group < x0, x1 | x1^-1*x0^-1*x1*x0 >
sage: A.meridians()
{0: [x1], 1: [x0], 2: [x1^-1*x0^-1*x1^-1]}
sage: G = A.fundamental_group(simplified=False)
sage: G.sorted_presentation()
Finitely presented group
< x0, x1, x2, x3 | x3^-1*x2^-1*x1^-1*x0^-1, x3^-1*x2^-1*x3*x0*x1*x0^-1,
                   x3^-1*x1^-1*x3*x0*x1*x0^-1*x2^-1*x0^-1*(x2*x0)^2*x1^-1*x0^-1,
                   x3^-1*x0^-1*x3*x0*x1*x0^-1*x2^-1*x0*x2*x0*x1^-1*x0^-1,
                   x2^-1*x0^-1*x2*x0, x1^-1*x0^-1*x1*x0 >
sage: A.meridians(simplified=False)
{0: [x1, x2], 1: [x0], 2: [x3]}
sage: A = H(y^2 + x*z, z, x)
sage: A.fundamental_group()
Finitely presented group < x0, x1 | (x1*x0)^2*(x1^-1*x0^-1)^2 >
sage: A = H(y^2 + x*z, z*x, y)
sage: A.fundamental_group()
Finitely presented group
< x0, x1, x2 | x2*x0*x1*x0^-1*x2^-1*x1^-1,
               x1*(x2*x0)^2*x2^-1*x1^-1*x0^-1*x2^-1*x0^-1 >

>>> from sage.all import *
>>> # needs sirocco
>>> H = ProjectivePlaneCurveArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = H._first_ngens(3)
>>> H(z).fundamental_group()
Finitely presented group <  |  >
>>> H(x*y).fundamental_group()
Finitely presented group < x |  >
>>> A = H(y**Integer(2) + x*z, y + x - z, x)
>>> A.fundamental_group().sorted_presentation()
Finitely presented group < x0, x1 | x1^-1*x0^-1*x1*x0 >
>>> A.meridians()
{0: [x1], 1: [x0], 2: [x1^-1*x0^-1*x1^-1]}
>>> G = A.fundamental_group(simplified=False)
>>> G.sorted_presentation()
Finitely presented group
< x0, x1, x2, x3 | x3^-1*x2^-1*x1^-1*x0^-1, x3^-1*x2^-1*x3*x0*x1*x0^-1,
                   x3^-1*x1^-1*x3*x0*x1*x0^-1*x2^-1*x0^-1*(x2*x0)^2*x1^-1*x0^-1,
                   x3^-1*x0^-1*x3*x0*x1*x0^-1*x2^-1*x0*x2*x0*x1^-1*x0^-1,
                   x2^-1*x0^-1*x2*x0, x1^-1*x0^-1*x1*x0 >
>>> A.meridians(simplified=False)
{0: [x1, x2], 1: [x0], 2: [x3]}
>>> A = H(y**Integer(2) + x*z, z, x)
>>> A.fundamental_group()
Finitely presented group < x0, x1 | (x1*x0)^2*(x1^-1*x0^-1)^2 >
>>> A = H(y**Integer(2) + x*z, z*x, y)
>>> A.fundamental_group()
Finitely presented group
< x0, x1, x2 | x2*x0*x1*x0^-1*x2^-1*x1^-1,
               x1*(x2*x0)^2*x2^-1*x1^-1*x0^-1*x2^-1*x0^-1 >
meridians(simplified=True)[source]#

Return the meridians of each irreducible component.

OUTPUT:

A dictionary which associates the index of each curve with its meridians, including the line at infinity if it can be computed

Note

This function requires the sirocco package to be installed and ProjectivePlaneCurveArrangements.fundamental_group() with the same options, where some examples are shown.

EXAMPLES:

sage: # needs sirocco
sage: H.<x, y, z> = ProjectivePlaneCurveArrangements(QQ)
sage: A = H(y^2 + x*z, y + x - z, x)
sage: A.fundamental_group().sorted_presentation()
Finitely presented group < x0, x1 | x1^-1*x0^-1*x1*x0 >
sage: A.meridians()
{0: [x1], 1: [x0], 2: [x1^-1*x0^-1*x1^-1]}
sage: A = H(y^2 + x*z, z, x)
sage: A.fundamental_group()
Finitely presented group < x0, x1 | (x1*x0)^2*(x1^-1*x0^-1)^2 >
sage: A.meridians()
{0: [x0, x1*x0*x1^-1], 1: [x0^-1*x1^-1*x0^-1], 2: [x1]}
sage: A = H(y^2 + x*z, z*x, y)
sage: A.fundamental_group()
Finitely presented group < x0, x1, x2 | x2*x0*x1*x0^-1*x2^-1*x1^-1,
                                        x1*(x2*x0)^2*x2^-1*x1^-1*x0^-1*x2^-1*x0^-1 >
sage: A.meridians()
{0: [x0, x2*x0*x2^-1], 1: [x2, x0^-1*x2^-1*x1^-1*x0^-1], 2: [x1]}

>>> from sage.all import *
>>> # needs sirocco
>>> H = ProjectivePlaneCurveArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = H._first_ngens(3)
>>> A = H(y**Integer(2) + x*z, y + x - z, x)
>>> A.fundamental_group().sorted_presentation()
Finitely presented group < x0, x1 | x1^-1*x0^-1*x1*x0 >
>>> A.meridians()
{0: [x1], 1: [x0], 2: [x1^-1*x0^-1*x1^-1]}
>>> A = H(y**Integer(2) + x*z, z, x)
>>> A.fundamental_group()
Finitely presented group < x0, x1 | (x1*x0)^2*(x1^-1*x0^-1)^2 >
>>> A.meridians()
{0: [x0, x1*x0*x1^-1], 1: [x0^-1*x1^-1*x0^-1], 2: [x1]}
>>> A = H(y**Integer(2) + x*z, z*x, y)
>>> A.fundamental_group()
Finitely presented group < x0, x1, x2 | x2*x0*x1*x0^-1*x2^-1*x1^-1,
                                        x1*(x2*x0)^2*x2^-1*x1^-1*x0^-1*x2^-1*x0^-1 >
>>> A.meridians()
{0: [x0, x2*x0*x2^-1], 1: [x2, x0^-1*x2^-1*x1^-1*x0^-1], 2: [x1]}
class sage.schemes.curves.plane_curve_arrangement.ProjectivePlaneCurveArrangements(base_ring, names=())[source]#

Bases: PlaneCurveArrangements

Projective curve arrangements.

INPUT:

  • base_ring – ring; the base ring

  • names – tuple of strings; the variable names

EXAMPLES:

sage: H.<x, y, z> = ProjectivePlaneCurveArrangements(QQ)
sage: H(x, y^2, x-z, y-z)
Arrangement (x, y^2, x - z, y - z) in Projective Space
of dimension 2 over Rational Field

>>> from sage.all import *
>>> H = ProjectivePlaneCurveArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = H._first_ngens(3)
>>> H(x, y**Integer(2), x-z, y-z)
Arrangement (x, y^2, x - z, y - z) in Projective Space
of dimension 2 over Rational Field
Element[source]#

alias of ProjectivePlaneCurveArrangementElement

ambient_space()[source]#

Return the ambient space.

EXAMPLES:

sage: L.<x, y, z> = ProjectivePlaneCurveArrangements(QQ)
sage: L.ambient_space()
Projective Space of dimension 2 over Rational Field

>>> from sage.all import *
>>> L = ProjectivePlaneCurveArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = L._first_ngens(3)
>>> L.ambient_space()
Projective Space of dimension 2 over Rational Field