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)

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()

Return the dimension of self.

OUTPUT: An integer.

EXAMPLES:

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

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

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

is_smooth(point=None)

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: # 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

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

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

class sage.schemes.toric.toric_subscheme.AlgebraicScheme_subscheme_toric(toric_variety, polynomials)

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)

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: 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)

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: 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

affine_patch(i)

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: 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

dimension()

Return the dimension of self.

OUTPUT: An integer. If self is empty, \(-1\) is returned.

EXAMPLES:

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

fan()

Return the fan of the ambient space.

OUTPUT: A fan.

EXAMPLES:

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

is_nondegenerate()

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: 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

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

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

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

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

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

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

It handles nonzero characteristic:

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

is_schon()

Check if self is schon (nondegenerate).

See is_nondegenerate() for further documentation.

EXAMPLES:

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

is_smooth(point=None)

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: # 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

Any sufficiently generic cubic hypersurface is smooth:

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

A more complicated example:

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

A smooth hypersurface in a compact singular toric variety:

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

neighborhood(point)

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: # 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]

A more complicated example:

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]