Subschemes of toric space¶

AUTHORS:

David Kohel (2005): initial version.
William Stein (2005): initial version.
Andrey Novoseltsev (2010-05-17): subschemes of toric varieties.

class sage.schemes.toric.toric_subscheme.AlgebraicScheme_subscheme_affine_toric(toric_variety, polynomials)[source]¶

Bases: AlgebraicScheme_subscheme_toric

Construct an algebraic subscheme of an affine toric variety.

Warning

You should not create objects of this class directly. The preferred method to construct such subschemes is to use subscheme() method of toric varieties.

INPUT:

toric_variety – ambient affine toric variety
polynomials – single polynomial, list, or ideal of defining polynomials in the coordinate ring of toric_variety

OUTPUT:

A algebraic subscheme of an affine toric variety.

dimension()[source]¶

Return the dimension of self.

OUTPUT: integer

EXAMPLES:

Sage

sage: # needs sage.libs.singular
sage: P1xP1.<s0,s1,t0,t1> = toric_varieties.P1xP1()
sage: P1 = P1xP1.subscheme(s0 - s1)
sage: P1.dimension()
1

Python

>>> from sage.all import *
>>> # needs sage.libs.singular
>>> P1xP1 = toric_varieties.P1xP1(names=('s0', 's1', 't0', 't1',)); (s0, s1, t0, t1,) = P1xP1._first_ngens(4)
>>> P1 = P1xP1.subscheme(s0 - s1)
>>> P1.dimension()
1

Sage Live

# needs sage.libs.singular
P1xP1.<s0,s1,t0,t1> = toric_varieties.P1xP1()
P1 = P1xP1.subscheme(s0 - s1)
P1.dimension()

A more complicated example where the ambient toric variety is not smooth:

Sage

sage: # needs sage.libs.singular
sage: X.<x,y> = toric_varieties.A2_Z2()
sage: X.is_smooth()
False
sage: Y = X.subscheme([x*y, x^2]); Y
Closed subscheme of 2-d affine toric variety defined by:
  x*y,
  x^2
sage: Y.dimension()
1

Python

>>> from sage.all import *
>>> # needs sage.libs.singular
>>> X = toric_varieties.A2_Z2(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> X.is_smooth()
False
>>> Y = X.subscheme([x*y, x**Integer(2)]); Y
Closed subscheme of 2-d affine toric variety defined by:
  x*y,
  x^2
>>> Y.dimension()
1

Sage Live

# needs sage.libs.singular
X.<x,y> = toric_varieties.A2_Z2()
X.is_smooth()
Y = X.subscheme([x*y, x^2]); Y
Y.dimension()

is_smooth(point=None)[source]¶

Test whether the algebraic subscheme is smooth.

INPUT:

point – a point or None (default); the point to test smoothness at

OUTPUT:

boolean; if no point was specified, returns whether the algebraic subscheme is smooth everywhere. Otherwise, smoothness at the specified point is tested.

EXAMPLES:

Sage

sage: # needs sage.libs.singular
sage: A2.<x,y> = toric_varieties.A2()
sage: cuspidal_curve = A2.subscheme([y^2 - x^3])
sage: cuspidal_curve
Closed subscheme of 2-d affine toric variety defined by:
  -x^3 + y^2
sage: cuspidal_curve.is_smooth([1,1])
True
sage: cuspidal_curve.is_smooth([0,0])
False
sage: cuspidal_curve.is_smooth()
False
sage: circle = A2.subscheme(x^2 + y^2 - 1)
sage: circle.is_smooth([1,0])
True
sage: circle.is_smooth()
True

Python

>>> from sage.all import *
>>> # needs sage.libs.singular
>>> A2 = toric_varieties.A2(names=('x', 'y',)); (x, y,) = A2._first_ngens(2)
>>> cuspidal_curve = A2.subscheme([y**Integer(2) - x**Integer(3)])
>>> cuspidal_curve
Closed subscheme of 2-d affine toric variety defined by:
  -x^3 + y^2
>>> cuspidal_curve.is_smooth([Integer(1),Integer(1)])
True
>>> cuspidal_curve.is_smooth([Integer(0),Integer(0)])
False
>>> cuspidal_curve.is_smooth()
False
>>> circle = A2.subscheme(x**Integer(2) + y**Integer(2) - Integer(1))
>>> circle.is_smooth([Integer(1),Integer(0)])
True
>>> circle.is_smooth()
True

Sage Live

# needs sage.libs.singular
A2.<x,y> = toric_varieties.A2()
cuspidal_curve = A2.subscheme([y^2 - x^3])
cuspidal_curve
cuspidal_curve.is_smooth([1,1])
cuspidal_curve.is_smooth([0,0])
cuspidal_curve.is_smooth()
circle = A2.subscheme(x^2 + y^2 - 1)
circle.is_smooth([1,0])
circle.is_smooth()

A more complicated example where the ambient toric variety is not smooth:

Sage

sage: # needs sage.libs.singular
sage: X.<x,y> = toric_varieties.A2_Z2()    # 2-d affine space mod Z/2
sage: X.is_smooth()
False
sage: Y = X.subscheme([x*y, x^2])   # (twice the x=0 curve) mod Z/2
sage: Y
Closed subscheme of 2-d affine toric variety defined by:
  x*y,
  x^2
sage: Y.dimension()   # Y is a Weil divisor but not Cartier
1
sage: Y.is_smooth()
True
sage: Y.is_smooth([0,0])
True

Python

>>> from sage.all import *
>>> # needs sage.libs.singular
>>> X = toric_varieties.A2_Z2(names=('x', 'y',)); (x, y,) = X._first_ngens(2)# 2-d affine space mod Z/2
>>> X.is_smooth()
False
>>> Y = X.subscheme([x*y, x**Integer(2)])   # (twice the x=0 curve) mod Z/2
>>> Y
Closed subscheme of 2-d affine toric variety defined by:
  x*y,
  x^2
>>> Y.dimension()   # Y is a Weil divisor but not Cartier
1
>>> Y.is_smooth()
True
>>> Y.is_smooth([Integer(0),Integer(0)])
True

Sage Live

# needs sage.libs.singular
X.<x,y> = toric_varieties.A2_Z2()    # 2-d affine space mod Z/2
X.is_smooth()
Y = X.subscheme([x*y, x^2])   # (twice the x=0 curve) mod Z/2
Y
Y.dimension()   # Y is a Weil divisor but not Cartier
Y.is_smooth()
Y.is_smooth([0,0])

class sage.schemes.toric.toric_subscheme.AlgebraicScheme_subscheme_toric(toric_variety, polynomials)[source]¶

Bases: AlgebraicScheme_subscheme

Construct an algebraic subscheme of a toric variety.

Warning

You should not create objects of this class directly. The preferred method to construct such subschemes is to use subscheme() method of toric varieties.

INPUT:

toric_variety – ambient toric variety
polynomials – single polynomial, list, or ideal of defining polynomials in the coordinate ring of toric_variety

OUTPUT: an algebraic subscheme of a toric variety.

affine_algebraic_patch(cone=None, names=None)[source]¶

Return the affine patch corresponding to cone as an affine algebraic scheme.

INPUT:

cone – a Cone \(\sigma\) of the fan. It can be omitted for an affine toric variety, in which case the single generating cone is used.

OUTPUT:

An affine algebraic subscheme corresponding to the patch \(\mathop{Spec}(\sigma^\vee \cap M)\) associated to the cone \(\sigma\).

See also affine_patch(), which expresses the patches as subvarieties of affine toric varieties instead.

REFERENCES:

David A. Cox, “The Homogeneous Coordinate Ring of a Toric Variety”, Lemma 2.2. arXiv alg-geom/9210008v2

EXAMPLES:

Sage

sage: P2.<x,y,z> = toric_varieties.P2()
sage: cone = P2.fan().generating_cone(0)
sage: V = P2.subscheme(x^3 + y^3 + z^3)
sage: V.affine_algebraic_patch(cone)
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  z0^3 + z1^3 + 1

sage: # needs fpylll sage.libs.singular
sage: cone = Cone([(0,1), (2,1)])
sage: A2Z2.<x,y> = AffineToricVariety(cone)
sage: A2Z2.affine_algebraic_patch()
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
  -z0*z1 + z2^2
sage: V = A2Z2.subscheme(x^2 + y^2 - 1)
sage: patch = V.affine_algebraic_patch();  patch
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
  -z0*z1 + z2^2,
  z0 + z1 - 1
sage: nbhd_patch = V.neighborhood([1,0]).affine_algebraic_patch();  nbhd_patch
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
  -z0*z1 + z2^2,
  z0 + z1 - 1
sage: nbhd_patch.embedding_center()
(0, 1, 0)

Python

>>> from sage.all import *
>>> P2 = toric_varieties.P2(names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> cone = P2.fan().generating_cone(Integer(0))
>>> V = P2.subscheme(x**Integer(3) + y**Integer(3) + z**Integer(3))
>>> V.affine_algebraic_patch(cone)
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  z0^3 + z1^3 + 1

>>> # needs fpylll sage.libs.singular
>>> cone = Cone([(Integer(0),Integer(1)), (Integer(2),Integer(1))])
>>> A2Z2 = AffineToricVariety(cone, names=('x', 'y',)); (x, y,) = A2Z2._first_ngens(2)
>>> A2Z2.affine_algebraic_patch()
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
  -z0*z1 + z2^2
>>> V = A2Z2.subscheme(x**Integer(2) + y**Integer(2) - Integer(1))
>>> patch = V.affine_algebraic_patch();  patch
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
  -z0*z1 + z2^2,
  z0 + z1 - 1
>>> nbhd_patch = V.neighborhood([Integer(1),Integer(0)]).affine_algebraic_patch();  nbhd_patch
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
  -z0*z1 + z2^2,
  z0 + z1 - 1
>>> nbhd_patch.embedding_center()
(0, 1, 0)

Sage Live

P2.<x,y,z> = toric_varieties.P2()
cone = P2.fan().generating_cone(0)
V = P2.subscheme(x^3 + y^3 + z^3)
V.affine_algebraic_patch(cone)
# needs fpylll sage.libs.singular
cone = Cone([(0,1), (2,1)])
A2Z2.<x,y> = AffineToricVariety(cone)
A2Z2.affine_algebraic_patch()
V = A2Z2.subscheme(x^2 + y^2 - 1)
patch = V.affine_algebraic_patch();  patch
nbhd_patch = V.neighborhood([1,0]).affine_algebraic_patch();  nbhd_patch
nbhd_patch.embedding_center()

Here we got two defining equations. The first one describes the singularity of the ambient space and the second is the pull-back of \(x^2+y^2-1\)

Sage

sage: lp = LatticePolytope([(1,0,0), (1,1,0), (1,1,1), (1,0,1), (-2,-1,-1)],
....:                      lattice=ToricLattice(3))
sage: X.<x,y,u,v,t> = CPRFanoToricVariety(Delta_polar=lp)
sage: Y = X.subscheme(x*v + y*u + t)
sage: cone = Cone([(1,0,0), (1,1,0), (1,1,1), (1,0,1)])
sage: Y.affine_algebraic_patch(cone)
Closed subscheme of Affine Space of dimension 4 over Rational Field defined by:
  z0*z2 - z1*z3,
  z1 + z3 + 1

Python

>>> from sage.all import *
>>> lp = LatticePolytope([(Integer(1),Integer(0),Integer(0)), (Integer(1),Integer(1),Integer(0)), (Integer(1),Integer(1),Integer(1)), (Integer(1),Integer(0),Integer(1)), (-Integer(2),-Integer(1),-Integer(1))],
...                      lattice=ToricLattice(Integer(3)))
>>> X = CPRFanoToricVariety(Delta_polar=lp, names=('x', 'y', 'u', 'v', 't',)); (x, y, u, v, t,) = X._first_ngens(5)
>>> Y = X.subscheme(x*v + y*u + t)
>>> cone = Cone([(Integer(1),Integer(0),Integer(0)), (Integer(1),Integer(1),Integer(0)), (Integer(1),Integer(1),Integer(1)), (Integer(1),Integer(0),Integer(1))])
>>> Y.affine_algebraic_patch(cone)
Closed subscheme of Affine Space of dimension 4 over Rational Field defined by:
  z0*z2 - z1*z3,
  z1 + z3 + 1

Sage Live

lp = LatticePolytope([(1,0,0), (1,1,0), (1,1,1), (1,0,1), (-2,-1,-1)],
                     lattice=ToricLattice(3))
X.<x,y,u,v,t> = CPRFanoToricVariety(Delta_polar=lp)
Y = X.subscheme(x*v + y*u + t)
cone = Cone([(1,0,0), (1,1,0), (1,1,1), (1,0,1)])
Y.affine_algebraic_patch(cone)

affine_patch(i)[source]¶

Return the i-th affine patch of self as an affine toric algebraic scheme.

INPUT:

i – integer; index of a generating cone of the fan of the ambient space of self

OUTPUT:

subscheme of an affine toric variety corresponding to the pull-back of self by the embedding morphism of the i-th affine patch of the ambient space of self.

The result is cached, so the i-th patch is always the same object in memory.

EXAMPLES:

Sage

sage: P1xP1 = toric_varieties.P1xP1()
sage: patch1 = P1xP1.affine_patch(1)
sage: patch1.embedding_morphism()
Scheme morphism:
  From: 2-d affine toric variety
  To:   2-d CPR-Fano toric variety covered by 4 affine patches
  Defn: Defined on coordinates by sending [t : x] to [1 : t : x : 1]
sage: P1xP1.inject_variables()
Defining s, t, x, y
sage: P1 = P1xP1.subscheme(x - y)
sage: subpatch = P1.affine_patch(1)
sage: subpatch
Closed subscheme of 2-d affine toric variety defined by:
  x - 1

Python

>>> from sage.all import *
>>> P1xP1 = toric_varieties.P1xP1()
>>> patch1 = P1xP1.affine_patch(Integer(1))
>>> patch1.embedding_morphism()
Scheme morphism:
  From: 2-d affine toric variety
  To:   2-d CPR-Fano toric variety covered by 4 affine patches
  Defn: Defined on coordinates by sending [t : x] to [1 : t : x : 1]
>>> P1xP1.inject_variables()
Defining s, t, x, y
>>> P1 = P1xP1.subscheme(x - y)
>>> subpatch = P1.affine_patch(Integer(1))
>>> subpatch
Closed subscheme of 2-d affine toric variety defined by:
  x - 1

Sage Live

P1xP1 = toric_varieties.P1xP1()
patch1 = P1xP1.affine_patch(1)
patch1.embedding_morphism()
P1xP1.inject_variables()
P1 = P1xP1.subscheme(x - y)
subpatch = P1.affine_patch(1)
subpatch

dimension()[source]¶

Return the dimension of self.

OUTPUT: integer; if self is empty, \(-1\) is returned

EXAMPLES:

Sage

sage: # needs sage.libs.singular
sage: P1xP1 = toric_varieties.P1xP1()
sage: P1xP1.inject_variables()
Defining s, t, x, y
sage: P1 = P1xP1.subscheme(s - t)
sage: P1.dimension()
1
sage: P1xP1.subscheme([s - t, (s-t)^2]).dimension()
1
sage: P1xP1.subscheme([s, t]).dimension()
-1

Python

>>> from sage.all import *
>>> # needs sage.libs.singular
>>> P1xP1 = toric_varieties.P1xP1()
>>> P1xP1.inject_variables()
Defining s, t, x, y
>>> P1 = P1xP1.subscheme(s - t)
>>> P1.dimension()
1
>>> P1xP1.subscheme([s - t, (s-t)**Integer(2)]).dimension()
1
>>> P1xP1.subscheme([s, t]).dimension()
-1

Sage Live

# needs sage.libs.singular
P1xP1 = toric_varieties.P1xP1()
P1xP1.inject_variables()
P1 = P1xP1.subscheme(s - t)
P1.dimension()
P1xP1.subscheme([s - t, (s-t)^2]).dimension()
P1xP1.subscheme([s, t]).dimension()

fan()[source]¶

Return the fan of the ambient space.

OUTPUT: a fan

EXAMPLES:

Sage

sage: P2.<x,y,z> = toric_varieties.P(2)
sage: E = P2.subscheme([x^2 + y^2 + z^2])
sage: E.fan()
Rational polyhedral fan in 2-d lattice N

Python

>>> from sage.all import *
>>> P2 = toric_varieties.P(Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> E = P2.subscheme([x**Integer(2) + y**Integer(2) + z**Integer(2)])
>>> E.fan()
Rational polyhedral fan in 2-d lattice N

Sage Live

P2.<x,y,z> = toric_varieties.P(2)
E = P2.subscheme([x^2 + y^2 + z^2])
E.fan()

is_nondegenerate()[source]¶

Check if self is nondegenerate.

OUTPUT:

Whether the variety is nondegenerate, that is, the intersection with every open torus orbit is smooth and transversal.

EXAMPLES:

Sage

sage: P2.<x,y,z> = toric_varieties.P2()
sage: P2.subscheme([x^3 + y^3 + z^3]).is_nondegenerate()
True
sage: P2.subscheme([x*y*z]).is_nondegenerate()
False
sage: X = P2.subscheme([(x-y)^2*(x+y) + x*y*z + z^3])
sage: X.is_smooth()
True
sage: X.is_nondegenerate()
False

Python

>>> from sage.all import *
>>> P2 = toric_varieties.P2(names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> P2.subscheme([x**Integer(3) + y**Integer(3) + z**Integer(3)]).is_nondegenerate()
True
>>> P2.subscheme([x*y*z]).is_nondegenerate()
False
>>> X = P2.subscheme([(x-y)**Integer(2)*(x+y) + x*y*z + z**Integer(3)])
>>> X.is_smooth()
True
>>> X.is_nondegenerate()
False

Sage Live

P2.<x,y,z> = toric_varieties.P2()
P2.subscheme([x^3 + y^3 + z^3]).is_nondegenerate()
P2.subscheme([x*y*z]).is_nondegenerate()
X = P2.subscheme([(x-y)^2*(x+y) + x*y*z + z^3])
X.is_smooth()
X.is_nondegenerate()

A K3 surface in \(\mathbf{P}^1 \times \mathbf{P}^1 \times \mathbf{P}^1\):

Sage

sage: diamond = lattice_polytope.cross_polytope(3)
sage: fan = FaceFan(diamond)
sage: P1xP1xP1 = ToricVariety(fan)
sage: z0, z1, z2, z3, z4, z5 = P1xP1xP1.gens()
sage: t = 5
sage: F = z0^2*z1^2*z2^2 + z1^2*z2^2*z3^2 + z0^2*z2^2*z4^2\
....: + z2^2*z3^2*z4^2 + t*z0*z1*z2*z3*z4*z5 + z0^2*z1^2*z5^2\
....: + z1^2*z3^2*z5^2 + z0^2*z4^2*z5^2 + z3^2*z4^2*z5^2
sage: X = P1xP1xP1.subscheme([F])
sage: X.is_smooth()
True
sage: X.is_nondegenerate()
False

Python

>>> from sage.all import *
>>> diamond = lattice_polytope.cross_polytope(Integer(3))
>>> fan = FaceFan(diamond)
>>> P1xP1xP1 = ToricVariety(fan)
>>> z0, z1, z2, z3, z4, z5 = P1xP1xP1.gens()
>>> t = Integer(5)
>>> F = z0**Integer(2)*z1**Integer(2)*z2**Integer(2) + z1**Integer(2)*z2**Integer(2)*z3**Integer(2) + z0**Integer(2)*z2**Integer(2)*z4**Integer(2)+ z2**Integer(2)*z3**Integer(2)*z4**Integer(2) + t*z0*z1*z2*z3*z4*z5 + z0**Integer(2)*z1**Integer(2)*z5**Integer(2)+ z1**Integer(2)*z3**Integer(2)*z5**Integer(2) + z0**Integer(2)*z4**Integer(2)*z5**Integer(2) + z3**Integer(2)*z4**Integer(2)*z5**Integer(2)
>>> X = P1xP1xP1.subscheme([F])
>>> X.is_smooth()
True
>>> X.is_nondegenerate()
False

Sage Live

diamond = lattice_polytope.cross_polytope(3)
fan = FaceFan(diamond)
P1xP1xP1 = ToricVariety(fan)
z0, z1, z2, z3, z4, z5 = P1xP1xP1.gens()
t = 5
F = z0^2*z1^2*z2^2 + z1^2*z2^2*z3^2 + z0^2*z2^2*z4^2\
+ z2^2*z3^2*z4^2 + t*z0*z1*z2*z3*z4*z5 + z0^2*z1^2*z5^2\
+ z1^2*z3^2*z5^2 + z0^2*z4^2*z5^2 + z3^2*z4^2*z5^2
X = P1xP1xP1.subscheme([F])
X.is_smooth()
X.is_nondegenerate()

Taking a random change of variables breaks the symmetry, but makes the surface nondegenerate:

Sage

sage: F1 = F.subs(z0=1*z0 + 1*z3, z3=1*z0 + 2*z3,
....:             z1=-2*z1 + -1*z4, z4=1*z1 + 2*z4,
....:             z2=-3*z2 + -1*z5, z5=-3*z2 + 2*z5)
sage: Y = P1xP1xP1.subscheme([F1])
sage: Y.is_smooth()
True
sage: Y.is_nondegenerate()
True

Python

>>> from sage.all import *
>>> F1 = F.subs(z0=Integer(1)*z0 + Integer(1)*z3, z3=Integer(1)*z0 + Integer(2)*z3,
...             z1=-Integer(2)*z1 + -Integer(1)*z4, z4=Integer(1)*z1 + Integer(2)*z4,
...             z2=-Integer(3)*z2 + -Integer(1)*z5, z5=-Integer(3)*z2 + Integer(2)*z5)
>>> Y = P1xP1xP1.subscheme([F1])
>>> Y.is_smooth()
True
>>> Y.is_nondegenerate()
True

Sage Live

F1 = F.subs(z0=1*z0 + 1*z3, z3=1*z0 + 2*z3,
            z1=-2*z1 + -1*z4, z4=1*z1 + 2*z4,
            z2=-3*z2 + -1*z5, z5=-3*z2 + 2*z5)
Y = P1xP1xP1.subscheme([F1])
Y.is_smooth()
Y.is_nondegenerate()

This example is from Hamm, arXiv 1106.1826v1. It addresses an issue raised at Issue #15239:

Sage

sage: X = toric_varieties.WP([1,4,2,3], names='z0 z1 z2 z3')
sage: X.inject_variables()
Defining z0, z1, z2, z3
sage: g0 = z1^3 + z2^6 + z3^4
sage: g = g0 - 2*z3^2*z0^6 + z2*z0^10 + z0^12
sage: Y = X.subscheme([g])
sage: Y.is_nondegenerate()
False

Python

>>> from sage.all import *
>>> X = toric_varieties.WP([Integer(1),Integer(4),Integer(2),Integer(3)], names='z0 z1 z2 z3')
>>> X.inject_variables()
Defining z0, z1, z2, z3
>>> g0 = z1**Integer(3) + z2**Integer(6) + z3**Integer(4)
>>> g = g0 - Integer(2)*z3**Integer(2)*z0**Integer(6) + z2*z0**Integer(10) + z0**Integer(12)
>>> Y = X.subscheme([g])
>>> Y.is_nondegenerate()
False

Sage Live

X = toric_varieties.WP([1,4,2,3], names='z0 z1 z2 z3')
X.inject_variables()
g0 = z1^3 + z2^6 + z3^4
g = g0 - 2*z3^2*z0^6 + z2*z0^10 + z0^12
Y = X.subscheme([g])
Y.is_nondegenerate()

It handles nonzero characteristic:

Sage

sage: P2.<x,y,z> = toric_varieties.P2()
sage: f = x^5 + 2*x*y^4 + y^5 - 2*y^3*z^2 + x*z^4 - 2*z^5
sage: P2.change_ring(GF(5)).subscheme([f]).is_nondegenerate()
True
sage: P2.change_ring(GF(7)).subscheme([f]).is_nondegenerate()
False

Python

>>> from sage.all import *
>>> P2 = toric_varieties.P2(names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> f = x**Integer(5) + Integer(2)*x*y**Integer(4) + y**Integer(5) - Integer(2)*y**Integer(3)*z**Integer(2) + x*z**Integer(4) - Integer(2)*z**Integer(5)
>>> P2.change_ring(GF(Integer(5))).subscheme([f]).is_nondegenerate()
True
>>> P2.change_ring(GF(Integer(7))).subscheme([f]).is_nondegenerate()
False

Sage Live

P2.<x,y,z> = toric_varieties.P2()
f = x^5 + 2*x*y^4 + y^5 - 2*y^3*z^2 + x*z^4 - 2*z^5
P2.change_ring(GF(5)).subscheme([f]).is_nondegenerate()
P2.change_ring(GF(7)).subscheme([f]).is_nondegenerate()

is_schon()[source]¶

Check if self is schon (nondegenerate).

See is_nondegenerate() for further documentation.

EXAMPLES:

Sage

sage: # needs sage.libs.singular
sage: P2.<x,y,z> = toric_varieties.P2()
sage: X = P2.subscheme([(x-y)^2*(x+y) + x*y*z + z^3])
sage: X.is_smooth()
True
sage: X.is_schon()
False

Python

>>> from sage.all import *
>>> # needs sage.libs.singular
>>> P2 = toric_varieties.P2(names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> X = P2.subscheme([(x-y)**Integer(2)*(x+y) + x*y*z + z**Integer(3)])
>>> X.is_smooth()
True
>>> X.is_schon()
False

Sage Live

# needs sage.libs.singular
P2.<x,y,z> = toric_varieties.P2()
X = P2.subscheme([(x-y)^2*(x+y) + x*y*z + z^3])
X.is_smooth()
X.is_schon()

is_smooth(point=None)[source]¶

Test whether the algebraic subscheme is smooth.

INPUT:

point – a point or None (default); the point to test smoothness at

OUTPUT:

boolean; if no point was specified, returns whether the algebraic subscheme is smooth everywhere. Otherwise, smoothness at the specified point is tested.

EXAMPLES:

Sage

sage: # needs sage.libs.singular
sage: P2.<x,y,z> = toric_varieties.P2()
sage: cuspidal_curve = P2.subscheme([y^2*z - x^3])
sage: cuspidal_curve
Closed subscheme of 2-d CPR-Fano toric variety covered by 3 affine patches defined by:
  -x^3 + y^2*z
sage: cuspidal_curve.is_smooth([1,1,1])
True
sage: cuspidal_curve.is_smooth([0,0,1])
False
sage: cuspidal_curve.is_smooth()
False

Python

>>> from sage.all import *
>>> # needs sage.libs.singular
>>> P2 = toric_varieties.P2(names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> cuspidal_curve = P2.subscheme([y**Integer(2)*z - x**Integer(3)])
>>> cuspidal_curve
Closed subscheme of 2-d CPR-Fano toric variety covered by 3 affine patches defined by:
  -x^3 + y^2*z
>>> cuspidal_curve.is_smooth([Integer(1),Integer(1),Integer(1)])
True
>>> cuspidal_curve.is_smooth([Integer(0),Integer(0),Integer(1)])
False
>>> cuspidal_curve.is_smooth()
False

Sage Live

# needs sage.libs.singular
P2.<x,y,z> = toric_varieties.P2()
cuspidal_curve = P2.subscheme([y^2*z - x^3])
cuspidal_curve
cuspidal_curve.is_smooth([1,1,1])
cuspidal_curve.is_smooth([0,0,1])
cuspidal_curve.is_smooth()

Any sufficiently generic cubic hypersurface is smooth:

Sage

sage: P2.subscheme([y^2*z-x^3+z^3+1/10*x*y*z]).is_smooth()                  # needs sage.libs.singular
True

Python

>>> from sage.all import *
>>> P2.subscheme([y**Integer(2)*z-x**Integer(3)+z**Integer(3)+Integer(1)/Integer(10)*x*y*z]).is_smooth()                  # needs sage.libs.singular
True

Sage Live

P2.subscheme([y^2*z-x^3+z^3+1/10*x*y*z]).is_smooth()                  # needs sage.libs.singular

A more complicated example:

Sage

sage: # needs sage.libs.singular
sage: dP6.<x0,x1,x2,x3,x4,x5> = toric_varieties.dP6()
sage: disjointP1s = dP6.subscheme(x0*x3)
sage: disjointP1s.is_smooth()
True
sage: intersectingP1s = dP6.subscheme(x0*x1)
sage: intersectingP1s.is_smooth()
False

Python

>>> from sage.all import *
>>> # needs sage.libs.singular
>>> dP6 = toric_varieties.dP6(names=('x0', 'x1', 'x2', 'x3', 'x4', 'x5',)); (x0, x1, x2, x3, x4, x5,) = dP6._first_ngens(6)
>>> disjointP1s = dP6.subscheme(x0*x3)
>>> disjointP1s.is_smooth()
True
>>> intersectingP1s = dP6.subscheme(x0*x1)
>>> intersectingP1s.is_smooth()
False

Sage Live

# needs sage.libs.singular
dP6.<x0,x1,x2,x3,x4,x5> = toric_varieties.dP6()
disjointP1s = dP6.subscheme(x0*x3)
disjointP1s.is_smooth()
intersectingP1s = dP6.subscheme(x0*x1)
intersectingP1s.is_smooth()

A smooth hypersurface in a compact singular toric variety:

Sage

sage: # needs sage.libs.singular
sage: lp = LatticePolytope([(1,0,0), (1,1,0), (1,1,1), (1,0,1), (-2,-1,-1)],
....:                      lattice=ToricLattice(3))
sage: X.<x,y,u,v,t> = CPRFanoToricVariety(Delta_polar=lp)
sage: Y = X.subscheme(x*v + y*u + t)
sage: cone = Cone([(1,0,0), (1,1,0), (1,1,1), (1,0,1)])
sage: Y.is_smooth()
True

Python

>>> from sage.all import *
>>> # needs sage.libs.singular
>>> lp = LatticePolytope([(Integer(1),Integer(0),Integer(0)), (Integer(1),Integer(1),Integer(0)), (Integer(1),Integer(1),Integer(1)), (Integer(1),Integer(0),Integer(1)), (-Integer(2),-Integer(1),-Integer(1))],
...                      lattice=ToricLattice(Integer(3)))
>>> X = CPRFanoToricVariety(Delta_polar=lp, names=('x', 'y', 'u', 'v', 't',)); (x, y, u, v, t,) = X._first_ngens(5)
>>> Y = X.subscheme(x*v + y*u + t)
>>> cone = Cone([(Integer(1),Integer(0),Integer(0)), (Integer(1),Integer(1),Integer(0)), (Integer(1),Integer(1),Integer(1)), (Integer(1),Integer(0),Integer(1))])
>>> Y.is_smooth()
True

Sage Live

# needs sage.libs.singular
lp = LatticePolytope([(1,0,0), (1,1,0), (1,1,1), (1,0,1), (-2,-1,-1)],
                     lattice=ToricLattice(3))
X.<x,y,u,v,t> = CPRFanoToricVariety(Delta_polar=lp)
Y = X.subscheme(x*v + y*u + t)
cone = Cone([(1,0,0), (1,1,0), (1,1,1), (1,0,1)])
Y.is_smooth()

neighborhood(point)[source]¶

Return a toric algebraic scheme isomorphic to neighborhood of the point.

INPUT:

point – a point of the toric algebraic scheme

OUTPUT:

An affine toric algebraic scheme (polynomial equations in an affine toric variety) with fixed embedding_morphism() and embedding_center().

EXAMPLES:

Sage

sage: # needs sage.libs.singular
sage: P.<x,y,z> = toric_varieties.P2()
sage: S = P.subscheme(x + 2*y + 3*z)
sage: s = S.point([0,-3,2]); s
[0 : -3 : 2]
sage: patch = S.neighborhood(s); patch
Closed subscheme of 2-d affine toric variety defined by:
  x + 2*y + 6
sage: patch.embedding_morphism()
Scheme morphism:
  From: Closed subscheme of 2-d affine toric variety defined by: x + 2*y + 6
  To:   Closed subscheme of 2-d CPR-Fano toric variety
        covered by 3 affine patches defined by: x + 2*y + 3*z
  Defn: Defined on coordinates by sending [x : y] to [-2*y - 6 : y : 2]
sage: patch.embedding_center()
[0 : -3]
sage: patch.embedding_morphism()(patch.embedding_center())
[0 : -3 : 2]

Python

>>> from sage.all import *
>>> # needs sage.libs.singular
>>> P = toric_varieties.P2(names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> S = P.subscheme(x + Integer(2)*y + Integer(3)*z)
>>> s = S.point([Integer(0),-Integer(3),Integer(2)]); s
[0 : -3 : 2]
>>> patch = S.neighborhood(s); patch
Closed subscheme of 2-d affine toric variety defined by:
  x + 2*y + 6
>>> patch.embedding_morphism()
Scheme morphism:
  From: Closed subscheme of 2-d affine toric variety defined by: x + 2*y + 6
  To:   Closed subscheme of 2-d CPR-Fano toric variety
        covered by 3 affine patches defined by: x + 2*y + 3*z
  Defn: Defined on coordinates by sending [x : y] to [-2*y - 6 : y : 2]
>>> patch.embedding_center()
[0 : -3]
>>> patch.embedding_morphism()(patch.embedding_center())
[0 : -3 : 2]

Sage Live

# needs sage.libs.singular
P.<x,y,z> = toric_varieties.P2()
S = P.subscheme(x + 2*y + 3*z)
s = S.point([0,-3,2]); s
patch = S.neighborhood(s); patch
patch.embedding_morphism()
patch.embedding_center()
patch.embedding_morphism()(patch.embedding_center())

A more complicated example:

Sage

sage: # needs sage.libs.singular
sage: dP6.<x0,x1,x2,x3,x4,x5> = toric_varieties.dP6()
sage: twoP1 = dP6.subscheme(x0*x3)
sage: patch = twoP1.neighborhood([0,1,2, 3,4,5]); patch
Closed subscheme of 2-d affine toric variety defined by:
  3*x0
sage: patch.embedding_morphism()
Scheme morphism:
  From: Closed subscheme of 2-d affine toric variety defined by: 3*x0
  To:   Closed subscheme of 2-d CPR-Fano toric variety
        covered by 6 affine patches defined by: x0*x3
  Defn: Defined on coordinates by sending [x0 : x1] to [0 : x1 : 2 : 3 : 4 : 5]
sage: patch.embedding_center()
[0 : 1]
sage: patch.embedding_morphism()(patch.embedding_center())
[0 : 1 : 2 : 3 : 4 : 5]

Python

>>> from sage.all import *
>>> # needs sage.libs.singular
>>> dP6 = toric_varieties.dP6(names=('x0', 'x1', 'x2', 'x3', 'x4', 'x5',)); (x0, x1, x2, x3, x4, x5,) = dP6._first_ngens(6)
>>> twoP1 = dP6.subscheme(x0*x3)
>>> patch = twoP1.neighborhood([Integer(0),Integer(1),Integer(2), Integer(3),Integer(4),Integer(5)]); patch
Closed subscheme of 2-d affine toric variety defined by:
  3*x0
>>> patch.embedding_morphism()
Scheme morphism:
  From: Closed subscheme of 2-d affine toric variety defined by: 3*x0
  To:   Closed subscheme of 2-d CPR-Fano toric variety
        covered by 6 affine patches defined by: x0*x3
  Defn: Defined on coordinates by sending [x0 : x1] to [0 : x1 : 2 : 3 : 4 : 5]
>>> patch.embedding_center()
[0 : 1]
>>> patch.embedding_morphism()(patch.embedding_center())
[0 : 1 : 2 : 3 : 4 : 5]

Sage Live

# needs sage.libs.singular
dP6.<x0,x1,x2,x3,x4,x5> = toric_varieties.dP6()
twoP1 = dP6.subscheme(x0*x3)
patch = twoP1.neighborhood([0,1,2, 3,4,5]); patch
patch.embedding_morphism()
patch.embedding_center()
patch.embedding_morphism()(patch.embedding_center())