Algebraic schemes

An algebraic scheme is defined by a set of polynomials in some suitable affine or projective coordinates. Possible ambient spaces are

Note that while projective spaces are of course toric varieties themselves, they are implemented differently in Sage due to efficiency considerations. You still can create a projective space as a toric variety if you wish.

In the future other ambient spaces, perhaps by means of gluing relations, may be intoduced.

Generally, polynomials p_0, p_1, \dots, p_n define an ideal I=\left<p_0, p_1, \dots, p_n\right>. In the projective and toric case, the polynomials (and, therefore, the ideal) must be homogeneous. The associated subscheme V(I) of the ambient space is, roughly speaking, the subset of the ambient space on which all polynomials vanish simultaneously.

Note

You should not construct algebraic scheme objects directly. Instead, use .subscheme() methods of ambient spaces. See below for examples.

EXAMPLES:

We first construct the ambient space, here the affine space \QQ^2:

sage: A2 = AffineSpace(2, QQ, 'x, y')
sage: A2.coordinate_ring().inject_variables()
Defining x, y

Now we can write polynomial equations in the variables x and y. For example, one equation cuts out a curve (a one-dimensional subscheme):

sage: V = A2.subscheme([x^2+y^2-1]); V
Closed subscheme of Affine Space of dimension 2
over Rational Field defined by:
  x^2 + y^2 - 1
sage: V.dimension()
1

Here is a more complicated example in a projective space:

sage: P3 = ProjectiveSpace(3, QQ, 'x')
sage: P3.inject_variables()
Defining x0, x1, x2, x3
sage: Q = matrix([[x0, x1, x2], [x1, x2, x3]]).minors(2); Q
[-x1^2 + x0*x2, -x1*x2 + x0*x3, -x2^2 + x1*x3]
sage: twisted_cubic = P3.subscheme(Q)
sage: twisted_cubic
Closed subscheme of Projective Space of dimension 3
over Rational Field defined by:
  -x1^2 + x0*x2,
  -x1*x2 + x0*x3,
  -x2^2 + x1*x3
sage: twisted_cubic.dimension()
1

Note that there are 3 equations in the 3-dimensional ambient space, yet the subscheme is 1-dimensional. One can show that it is not possible to eliminate any of the equations, that is, the twisted cubic is not a complete intersection of two polynomial equations.

Let us look at one affine patch, for example the one where x_0=1

sage: patch = twisted_cubic.affine_patch(0)
sage: patch
Closed subscheme of Affine Space of dimension 3
over Rational Field defined by:
  -x0^2 + x1,
  -x0*x1 + x2,
  -x1^2 + x0*x2
sage: patch.projective_embedding()
Scheme morphism:
  From: Closed subscheme of Affine Space of dimension 3
  over Rational Field defined by:
  -x0^2 + x1,
  -x0*x1 + x2,
  -x1^2 + x0*x2
  To:   Closed subscheme of Projective Space of dimension 3
  over Rational Field defined by:
  -x1^2 + x0*x2,
  -x1*x2 + x0*x3,
  -x2^2 + x1*x3
  Defn: Defined on coordinates by sending (x0, x1, x2) to
        (1 : x0bar : x1bar : x2bar)

AUTHORS:

  • David Kohel (2005): initial version.
  • William Stein (2005): initial version.
  • Andrey Novoseltsev (2010-05-17): subschemes of toric varieties.
  • Volker Braun (2010-12-24): documentation of schemes and refactoring.
class sage.schemes.generic.algebraic_scheme.AlgebraicScheme(A)

Bases: sage.schemes.generic.scheme.Scheme

An algebraic scheme presented as a subscheme in an ambient space.

This is the base class for all algebraic schemes, that is, schemes defined by equations in affine, projective, or toric ambient spaces.

ambient_space()

Return the ambient space of this algebraic scheme.

EXAMPLES:

sage: A.<x, y> = AffineSpace(2, GF(5))
sage: S = A.subscheme([])
sage: S.ambient_space()
Affine Space of dimension 2 over Finite Field of size 5

sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: S = P.subscheme([x-y, x-z])
sage: S.ambient_space() is P
True
coordinate_ring()

Return the coordinate ring of this algebraic scheme. The result is cached.

EXAMPLES:

sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: S = P.subscheme([x-y, x-z])
sage: S.coordinate_ring()
Quotient of Multivariate Polynomial Ring in x, y, z over Integer Ring by the ideal (x - y, x - z)
is_projective()

Return True if self is presented as a subscheme of an ambient projective space.

EXAMPLES:

sage: PP.<x,y,z,w> = ProjectiveSpace(3,QQ)
sage: f = x^3 + y^3 + z^3 + w^3
sage: R = f.parent()
sage: I = [f] + [f.derivative(zz) for zz in PP.gens()]
sage: V = PP.subscheme(I)
sage: V.is_projective()
True
sage: AA.<x,y,z,w> = AffineSpace(4,QQ)
sage: V = AA.subscheme(I)
sage: V.is_projective()
False
ngens()

Return the number of generators of the ambient space of this algebraic scheme.

EXAMPLES:

sage: A.<x, y> = AffineSpace(2, GF(5))
sage: S = A.subscheme([])
sage: S.ngens()
2

sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: S = P.subscheme([x-y, x-z])
sage: P.ngens()
3
class sage.schemes.generic.algebraic_scheme.AlgebraicScheme_quasi(X, Y)

Bases: sage.schemes.generic.algebraic_scheme.AlgebraicScheme

The quasi-affine or quasi-projective scheme X - Y, where X and Y are both closed subschemes of a common ambient affine or projective space.

X()

Return the scheme X such that self is represented as X - Y.

EXAMPLES:

sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: S = P.subscheme([])
sage: T = P.subscheme([x-y])
sage: U = T.complement(S)
sage: U.X() is S
True
Y()

Return the scheme Y such that self is represented as X - Y.

EXAMPLES:

sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: S = P.subscheme([])
sage: T = P.subscheme([x-y])
sage: U = T.complement(S)
sage: U.Y() is T
True
rational_points(F=None, bound=0)

Return the set of rational points on this algebraic scheme over the field F.

EXAMPLES:

sage: A.<x, y> = AffineSpace(2, GF(7))
sage: S = A.subscheme([x^2-y])
sage: T = A.subscheme([x-y])
sage: U = T.complement(S)
sage: U.rational_points()
[(2, 4), (3, 2), (4, 2), (5, 4), (6, 1)]
sage: U.rational_points(GF(7^2, 'b'))
[(2, 4), (3, 2), (4, 2), (5, 4), (6, 1), (b, b + 4), (b + 1, 3*b + 5), (b + 2, 5*b + 1),
(b + 3, 6), (b + 4, 2*b + 6), (b + 5, 4*b + 1), (b + 6, 6*b + 5), (2*b, 4*b + 2),
(2*b + 1, b + 3), (2*b + 2, 5*b + 6), (2*b + 3, 2*b + 4), (2*b + 4, 6*b + 4),
(2*b + 5, 3*b + 6), (2*b + 6, 3), (3*b, 2*b + 1), (3*b + 1, b + 2), (3*b + 2, 5),
(3*b + 3, 6*b + 3), (3*b + 4, 5*b + 3), (3*b + 5, 4*b + 5), (3*b + 6, 3*b + 2),
(4*b, 2*b + 1), (4*b + 1, 3*b + 2), (4*b + 2, 4*b + 5), (4*b + 3, 5*b + 3),
(4*b + 4, 6*b + 3), (4*b + 5, 5), (4*b + 6, b + 2), (5*b, 4*b + 2), (5*b + 1, 3),
(5*b + 2, 3*b + 6), (5*b + 3, 6*b + 4), (5*b + 4, 2*b + 4), (5*b + 5, 5*b + 6),
(5*b + 6, b + 3), (6*b, b + 4), (6*b + 1, 6*b + 5), (6*b + 2, 4*b + 1), (6*b + 3, 2*b + 6),
(6*b + 4, 6), (6*b + 5, 5*b + 1), (6*b + 6, 3*b + 5)]
class sage.schemes.generic.algebraic_scheme.AlgebraicScheme_subscheme(A, polynomials)

Bases: sage.schemes.generic.algebraic_scheme.AlgebraicScheme

An algebraic scheme presented as a closed subscheme is defined by explicit polynomial equations. This is as opposed to a general scheme, which could, e.g., be the Neron model of some object, and for which we do not want to give explicit equations.

INPUT:

  • A - ambient space (e.g. affine or projective n-space)

  • polynomials - single polynomial, ideal or iterable of defining

    polynomials; in any case polynomials must belong to the coordinate ring of the ambient space and define valid polynomial functions (e.g. they should be homogeneous in the case of a projective space)

OUTPUT:

  • algebraic scheme

EXAMPLES:

sage: from sage.schemes.generic.algebraic_scheme import AlgebraicScheme_subscheme
sage: P.<x, y, z> = ProjectiveSpace(2, QQ)
sage: P.subscheme([x^2-y*z])
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
  x^2 - y*z
sage: AlgebraicScheme_subscheme(P, [x^2-y*z])
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
  x^2 - y*z
base_extend(R)

Return the base change to the ring R of this scheme.

EXAMPLES:

sage: P.<x, y, z> = ProjectiveSpace(2, GF(11))
sage: S = P.subscheme([x^2-y*z])
sage: S.base_extend(GF(11^2, 'b'))
Closed subscheme of Projective Space of dimension 2 over Finite Field in b of size 11^2 defined by:
  x^2 - y*z
sage: S.base_extend(ZZ)
Traceback (most recent call last):
...
ValueError: no natural map from the base ring (=Finite Field of size 11) to R (=Integer Ring)!
complement(other=None)

Return the scheme-theoretic complement other - self, where self and other are both closed algebraic subschemes of the same ambient space.

If other is unspecified, it is taken to be the ambient space of self.

EXAMPLES:

sage: A.<x, y, z> = AffineSpace(3, ZZ)
sage: X = A.subscheme([x+y-z])
sage: Y = A.subscheme([x-y+z])
sage: Y.complement(X)
Quasi-affine subscheme X - Y of Affine Space of dimension 3 over Integer Ring, where X is defined by:
  x + y - z
and Y is defined by:
  x - y + z
sage: Y.complement()
Quasi-affine subscheme X - Y of Affine Space of dimension 3 over Integer Ring, where X is defined by:
  (no polynomials)
and Y is defined by:
  x - y + z
sage: P.<x, y, z> = ProjectiveSpace(2, QQ)
sage: X = P.subscheme([x^2+y^2+z^2])
sage: Y = P.subscheme([x*y+y*z+z*x])
sage: Y.complement(X)
Quasi-projective subscheme X - Y of Projective Space of dimension 2 over Rational Field, where X is defined by:
  x^2 + y^2 + z^2
and Y is defined by:
  x*y + x*z + y*z
sage: Y.complement(P)
Quasi-projective subscheme X - Y of Projective Space of dimension 2 over Rational Field, where X is defined by:
  (no polynomials)
and Y is defined by:
  x*y + x*z + y*z
defining_ideal()

Return the ideal that defines this scheme as a subscheme of its ambient space.

EXAMPLES:

sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: S = P.subscheme([x^2-y*z, x^3+z^3])
sage: S.defining_ideal()
Ideal (x^2 - y*z, x^3 + z^3) of Multivariate Polynomial Ring in x, y, z over Integer Ring
defining_polynomials()

Return the polynomials that define this scheme as a subscheme of its ambient space.

EXAMPLES:

sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: S = P.subscheme([x^2-y*z, x^3+z^3])
sage: S.defining_polynomials()
(x^2 - y*z, x^3 + z^3)
intersection(other)

Return the scheme-theoretic intersection of self and other in their common ambient space.

EXAMPLES:

sage: A.<x, y> = AffineSpace(2, ZZ)
sage: X = A.subscheme([x^2-y])
sage: Y = A.subscheme([y])
sage: X.intersection(Y)
Closed subscheme of Affine Space of dimension 2 over Integer Ring defined by:
  x^2 - y,
  y
irreducible_components()

Return the irreducible components of this algebraic scheme, as subschemes of the same ambient space.

OUTPUT: an immutable sequence of irreducible subschemes of the ambient space of this scheme

The components are cached.

EXAMPLES:

We define what is clearly a union of four hypersurfaces in \P^4_{\QQ} then find the irreducible components.

sage: PP.<x,y,z,w,v> = ProjectiveSpace(4,QQ)
sage: V = PP.subscheme( (x^2 - y^2 - z^2)*(w^5 -  2*v^2*z^3)* w * (v^3 - x^2*z) )
sage: V.irreducible_components()
[
Closed subscheme of Projective Space of dimension 4 over Rational Field defined by:
w,
Closed subscheme of Projective Space of dimension 4 over Rational Field defined by:
x^2 - y^2 - z^2,
Closed subscheme of Projective Space of dimension 4 over Rational Field defined by:
x^2*z - v^3,
Closed subscheme of Projective Space of dimension 4 over Rational Field defined by:
w^5 - 2*z^3*v^2
]

We verify that the irrelevant ideal isn’t accidently returned (see trac 6920):

sage: PP.<x,y,z,w> = ProjectiveSpace(3,QQ)
sage: f = x^3 + y^3 + z^3 + w^3
sage: R = f.parent()
sage: I = [f] + [f.derivative(zz) for zz in PP.gens()]
sage: V = PP.subscheme(I)
sage: V.irreducible_components()
[
<BLANKLINE>
]

The same polynomial as above defines a scheme with a nontrivial irreducible component in affine space (instead of the empty scheme as above):

sage: AA.<x,y,z,w> = AffineSpace(4,QQ)
sage: V = AA.subscheme(I)
sage: V.irreducible_components()
[
Closed subscheme of Affine Space of dimension 4 over Rational Field defined by:
  w,
  z,
  y,
  x
]            
rational_points(F=None, bound=0)

EXAMPLES:

One can enumerate points up to a given bound on a projective scheme over the rationals.

sage: E = EllipticCurve('37a')
sage: E.rational_points(bound=8)
[(-1 : -1 : 1), (-1 : 0 : 1), (0 : -1 : 1), (0 : 0 : 1), (0 : 1 : 0), (1/4 : -5/8 : 1), (1/4 : -3/8 : 1), (1 : -1 : 1), (1 : 0 : 1), (2 : -3 : 1), (2 : 2 : 1)]

For a small finite field, the complete set of points can be enumerated.

sage: Etilde = E.base_extend(GF(3))
sage: Etilde.rational_points()
[(0 : 0 : 1), (0 : 1 : 0), (0 : 2 : 1), (1 : 0 : 1), (1 : 2 : 1), (2 : 0 : 1), (2 : 2 : 1)]

The class of hyperelliptic curves does not (yet) support desingularization of the places at infinity into two points.

sage: FF = FiniteField(7)
sage: P.<x> = PolynomialRing(FiniteField(7))
sage: C = HyperellipticCurve(x^8+x+1)
sage: C.rational_points()
[(0 : 1 : 0), (0 : 1 : 1), (0 : 6 : 1), (2 : 0 : 1), (4 : 0 : 1), (6 : 1 : 1), (6 : 6 : 1)]

TODO:

  1. The above algorithms enumerate all projective points and test whether they lie on the scheme; Implement a more naive sieve at least for covers of the projective line.
  2. Implement Stoll’s model in weighted projective space to resolve singularities and find two points (1 : 1 : 0) and (-1 : 1 : 0) at infinity.
reduce()

Return the corresponding reduced algebraic space associated to this scheme.

EXAMPLES: First we construct the union of a doubled and tripled line in the affine plane over \QQ.

sage: A.<x,y> = AffineSpace(2, QQ)
sage: X = A.subscheme([(x-1)^2*(x-y)^3]); X
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  x^5 - 3*x^4*y + 3*x^3*y^2 - x^2*y^3 - 2*x^4 + 6*x^3*y - 6*x^2*y^2 + 2*x*y^3 + x^3 - 3*x^2*y + 3*x*y^2 - y^3
sage: X.dimension()
1

Then we compute the corresponding reduced scheme.

sage: Y = X.reduce(); Y
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  x^2 - x*y - x + y

Finally, we verify that the reduced scheme Y is the union of those two lines.

sage: L1 = A.subscheme([x-1]); L1
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  x - 1
sage: L2 = A.subscheme([x-y]); L2
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  x - y
sage: W = L1.union(L2); W             # taken in ambient space
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  x^2 - x*y - x + y
sage: Y == W
True
union(other)

Return the scheme-theoretic union of self and other in their common ambient space.

EXAMPLES: We construct the union of a line and a tripled-point on the line.

sage: A.<x,y> = AffineSpace(2, QQ)
sage: I = ideal([x,y])^3
sage: P = A.subscheme(I)
sage: L = A.subscheme([y-1])
sage: S = L.union(P); S
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
y^4 - y^3,
x*y^3 - x*y^2,
x^2*y^2 - x^2*y,
x^3*y - x^3
sage: S.dimension()
1
sage: S.reduce()
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
y^2 - y,
x*y - x

We can also use the notation “+” for the union:

sage: A.subscheme([x]) + A.subscheme([y^2 - (x^3+1)])
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
-x^4 + x*y^2 - x

Saving and loading:

sage: loads(S.dumps()) == S
True
class sage.schemes.generic.algebraic_scheme.AlgebraicScheme_subscheme_affine(A, polynomials)

Bases: sage.schemes.generic.algebraic_scheme.AlgebraicScheme_subscheme

dimension()

EXAMPLES:

sage: A.<x,y> = AffineSpace(2, QQ)
sage: A.subscheme([]).dimension()
2
sage: A.subscheme([x]).dimension()
1
sage: A.subscheme([x^5]).dimension()
1
sage: A.subscheme([x^2 + y^2 - 1]).dimension()
1
sage: A.subscheme([x*(x-1), y*(y-1)]).dimension()
0

Something less obvious

sage: A.<x,y,z,w> = AffineSpace(4, QQ)
sage: X = A.subscheme([x^2, x^2*y^2 + z^2, z^2 - w^2, 10*x^2 + w^2 - z^2])
sage: X
Closed subscheme of Affine Space of dimension 4 over Rational Field defined by:
  x^2,
  x^2*y^2 + z^2,
  z^2 - w^2,
  10*x^2 - z^2 + w^2
sage: X.dimension()
1
projective_embedding(i=None, X=None)

Returns a morphism from this affine scheme into an ambient projective space of the same dimension.

INPUT:

  • i - integer (default: dimension of self = last coordinate) determines which projective embedding to compute. The embedding is that which has a 1 in the i-th coordinate, numbered from 0.
  • X - (default: None) projective scheme, i.e., codomain of morphism; this is constructed if it is not given.

EXAMPLES:

sage: A.<x, y, z> = AffineSpace(3, ZZ)
sage: S = A.subscheme([x*y-z])
sage: S.projective_embedding()
Scheme morphism:                  
  From: Closed subscheme of Affine Space of dimension 3 over Integer Ring defined by:
  x*y - z
  To:   Closed subscheme of Projective Space of dimension 3 over Integer Ring defined by:
  x0*x1 - x2*x3
  Defn: Defined on coordinates by sending (x, y, z) to
        (xbar : ybar : zbar : 1)
class sage.schemes.generic.algebraic_scheme.AlgebraicScheme_subscheme_projective(A, polynomials)

Bases: sage.schemes.generic.algebraic_scheme.AlgebraicScheme_subscheme

affine_patch(i)

Return the i^{th} affine patch of this projective scheme. This is the intersection with this i^{th} affine patch of its ambient space.

INPUT:

  • i – integer between 0 and dimension of self, inclusive.

OUTPUT:

An affine scheme with fixed projective_embedding() map.

EXAMPLES:

sage: PP = ProjectiveSpace(2, QQ, names='X,Y,Z')
sage: X,Y,Z = PP.gens()
sage: C = PP.subscheme(X^3*Y + Y^3*Z + Z^3*X)
sage: U = C.affine_patch(0)
sage: U
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x0^3*x1 + x1^3 + x0
sage: U.projective_embedding()
Scheme morphism:
  From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  x0^3*x1 + x1^3 + x0
  To:   Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
  X^3*Y + Y^3*Z + X*Z^3
  Defn: Defined on coordinates by sending (x0, x1) to
        (1 : x0bar : x1bar)
dimension()

EXAMPLES:

sage: A.<x,y> = AffineSpace(2, QQ)
sage: A.subscheme([]).dimension()
2
sage: A.subscheme([x]).dimension()
1
sage: A.subscheme([x^5]).dimension()
1
sage: A.subscheme([x^2 + y^2 - 1]).dimension()
1
sage: A.subscheme([x*(x-1), y*(y-1)]).dimension()
0

Something less obvious

sage: A.<x,y,z,w> = AffineSpace(4, QQ)
sage: X = A.subscheme([x^2, x^2*y^2 + z^2, z^2 - w^2, 10*x^2 + w^2 - z^2])
sage: X
Closed subscheme of Affine Space of dimension 4 over Rational Field defined by:
  x^2,
  x^2*y^2 + z^2,
  z^2 - w^2,
  10*x^2 - z^2 + w^2
sage: X.dimension()
1
class sage.schemes.generic.algebraic_scheme.AlgebraicScheme_subscheme_toric(toric_variety, polynomials)

Bases: sage.schemes.generic.algebraic_scheme.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:

TESTS:

sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan, "x s y t")
sage: P1xP1.inject_variables()
Defining x, s, y, t
sage: import sage.schemes.generic.algebraic_scheme as SCM
sage: X = SCM.AlgebraicScheme_subscheme_toric(
...         P1xP1, [x*s + y*t, x^3+y^3])
sage: X
Closed subscheme of 2-d toric variety
covered by 4 affine patches defined by:
  x*s + y*t,
  x^3 + y^3

A better way to construct the same scheme as above:

sage: P1xP1.subscheme([x*s + y*t, x^3+y^3])
Closed subscheme of 2-d toric variety
covered by 4 affine patches defined by:
  x*s + y*t,
  x^3 + y^3
affine_patch(i)

Return the i-th affine patch of self.

INPUT:

  • i – integer, index of a generating cone of the fan of the ambient space of self.

OUTPUT:

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

EXAMPLES:

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

Return the dimension of self.

Note

Currently the dimension of subschemes of toric varieties can be returned only if it was somehow set before.

OUTPUT:

  • integer.

EXAMPLES:

sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1.inject_variables()
Defining z0, z1, z2, z3
sage: P1 = P1xP1.subscheme(z0-z2)
sage: P1.dimension()
Traceback (most recent call last):
...
NotImplementedError:
cannot compute dimension of this scheme!
embedding_morphism()

Return the default embedding morphism of self.

Such a morphism is always defined for an affine patch of a subscheme of a toric variety (which is a subscheme of a toric variety itself).

OUTPUT:

  • scheme morphism if the default embedding morphism was defined for self, otherwise a ValueError exception is raised.

EXAMPLES:

sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan, "x s y t")
sage: patch1 = P1xP1.affine_patch(1)
sage: patch1.embedding_morphism()
Scheme morphism:
  From: 2-d affine toric variety
  To:   2-d toric variety covered by 4 affine patches
  Defn: Defined on coordinates by sending [y : t] to
        [1 : 1 : y : t]
sage: P1xP1.inject_variables()
Defining x, s, y, t
sage: P1 = P1xP1.subscheme(x-y)
sage: P1.embedding_morphism()
Traceback (most recent call last):
...
ValueError: no default embedding was defined
for this subscheme of a toric variety!
sage: subpatch = P1.affine_patch(1)
sage: subpatch
Closed subscheme of 2-d affine toric variety defined by:
  -y + 1
sage: subpatch.embedding_morphism()
Scheme morphism:
  From: Closed subscheme of 2-d affine toric variety defined by:
  -y + 1
  To:   Closed subscheme of 2-d toric variety
  covered by 4 affine patches defined by:
  x - y
  Defn: Defined on coordinates by sending [y : t] to
        [1 : 1 : 1 : tbar]
sage.schemes.generic.algebraic_scheme.is_AlgebraicScheme(x)

Test whether x is an algebraic scheme.

INPUT:

  • x – anything.

OUTPUT:

Boolean. Whether x is an an algebraic scheme, that is, a subscheme of an ambient space over a ring defined by polynomial equations.

EXAMPLES:

sage: A2 = AffineSpace(2, QQ, 'x, y')
sage: A2.coordinate_ring().inject_variables()
Defining x, y
sage: V = A2.subscheme([x^2+y^2]); V
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  x^2 + y^2
sage: from sage.schemes.generic.algebraic_scheme import is_AlgebraicScheme
sage: is_AlgebraicScheme(V)
True

Affine space is itself not an algebraic scheme, though the closed subscheme defined by no equations is:

sage: from sage.schemes.generic.algebraic_scheme import is_AlgebraicScheme
sage: is_AlgebraicScheme(AffineSpace(10, QQ))
False
sage: V = AffineSpace(10, QQ).subscheme([]); V
Closed subscheme of Affine Space of dimension 10 over Rational Field defined by:
  (no polynomials)
sage: is_AlgebraicScheme(V)
True

We create a more complicated closed subscheme:

sage: A, x = AffineSpace(10, QQ).objgens()
sage: X = A.subscheme([sum(x)]); X
Closed subscheme of Affine Space of dimension 10 over Rational Field defined by:
x0 + x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9
sage: is_AlgebraicScheme(X)
True
sage: is_AlgebraicScheme(QQ)
False
sage: S = Spec(QQ)
sage: is_AlgebraicScheme(S)
False

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