An algebraic scheme is defined by a set of polynomials in some suitable affine or projective coordinates. Possible ambient spaces are
- Affine spaces (AffineSpace),
- Projective spaces (ProjectiveSpace), or
- Toric varieties (ToricVariety).
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 following, we call the corresponding subschemes affine algebraic schemes, projective algebraic schemes, or toric algebraic schemes. 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.
Warning
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.embedding_morphism()
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 : x0 : x1 : x2)
AUTHORS:
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.
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
Return the coordinate ring of this algebraic scheme. The result is cached.
OUTPUT:
The coordinate ring. Usually a polynomial ring, or a quotient thereof.
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)
Return the distinguished point, if there is any.
If the scheme \(Y\) was constructed as a neighbourhood of a point \(p \in X\), then embedding_morphism() returns a local isomorphism \(f:Y\to X\) around the preimage point \(f^{-1}(p)\). The latter is returned by embedding_center().
OUTPUT:
A point of self. Raises AttributeError if there is no distinguished point, depending on how self was constructed.
EXAMPLES:
sage: P3.<w,x,y,z> = ProjectiveSpace(QQ,3)
sage: X = P3.subscheme( (w^2-x^2)*(y^2-z^2) )
sage: p = [1,-1,3,4]
sage: nbhd = X.neighborhood(p); nbhd
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
x0^2*x2^2 - x1^2*x2^2 + 6*x0^2*x2 - 6*x1^2*x2 + 2*x0*x2^2 +
2*x1*x2^2 - 7*x0^2 + 7*x1^2 + 12*x0*x2 + 12*x1*x2 - 14*x0 - 14*x1
sage: nbhd.embedding_center()
(0, 0, 0)
sage: nbhd.embedding_morphism()(nbhd.embedding_center())
(1/4 : -1/4 : 3/4 : 1)
sage: nbhd.embedding_morphism()
Scheme morphism:
From: Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
x0^2*x2^2 - x1^2*x2^2 + 6*x0^2*x2 - 6*x1^2*x2 + 2*x0*x2^2 +
2*x1*x2^2 - 7*x0^2 + 7*x1^2 + 12*x0*x2 + 12*x1*x2 - 14*x0 - 14*x1
To: Closed subscheme of Projective Space of dimension 3 over Rational Field defined by:
w^2*y^2 - x^2*y^2 - w^2*z^2 + x^2*z^2
Defn: Defined on coordinates by sending (x0, x1, x2) to
(x0 + 1 : x1 - 1 : x2 + 3 : 4)
Return the default embedding morphism of self.
If the scheme \(Y\) was constructed as a neighbourhood of a point \(p \in X\), then embedding_morphism() returns a local isomorphism \(f:Y\to X\) around the preimage point \(f^{-1}(p)\). The latter is returned by embedding_center().
If the algebraic scheme \(Y\) was not constructed as a neighbourhood of a point, then the embedding in its ambient_space() is returned.
OUTPUT:
A scheme morphism whose domain() is self.
EXAMPLES:
sage: A2.<x,y> = AffineSpace(QQ,2)
sage: C = A2.subscheme(x^2+y^2-1)
sage: C.embedding_morphism()
Scheme morphism:
From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x^2 + y^2 - 1
To: Affine Space of dimension 2 over Rational Field
Defn: Defined on coordinates by sending (x, y) to
(x, y)
sage: P1xP1.<x,y,u,v> = toric_varieties.P1xP1()
sage: P1 = P1xP1.subscheme(x-y)
sage: P1.embedding_morphism()
Scheme morphism:
From: Closed subscheme of 2-d CPR-Fano toric variety covered
by 4 affine patches defined by:
x - y
To: 2-d CPR-Fano toric variety covered by 4 affine patches
Defn: Defined on coordinates by sending [x : y : u : v] to
[y : y : u : v]
So far, the embedding was just in the own ambient space. Now a bit more interesting examples:
sage: P2.<x,y,z> = ProjectiveSpace(QQ,2)
sage: X = P2.subscheme((x^2-y^2)*z)
sage: p = (1,1,0)
sage: nbhd = X.neighborhood(p)
sage: nbhd
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
-x0^2*x1 - 2*x0*x1
Note that \(p=(1,1,0)\) is a singular point of \(X\). So the neighborhood of \(p\) is not just affine space. The :meth:neighborhood` method returns a presentation of the neighborhood as a subscheme of an auxiliary 2-dimensional affine space:
sage: nbhd.ambient_space()
Affine Space of dimension 2 over Rational Field
But its embedding_morphism() is not into this auxiliary affine space, but the original subscheme \(X\):
sage: nbhd.embedding_morphism()
Scheme morphism:
From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
-x0^2*x1 - 2*x0*x1
To: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x^2*z - y^2*z
Defn: Defined on coordinates by sending (x0, x1) to
(1 : x0 + 1 : x1)
A couple more examples:
sage: patch1 = P1xP1.affine_patch(1)
sage: patch1
2-d affine toric variety
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 [y : u] to
[1 : y : u : 1]
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 CPR-Fano toric variety covered
by 4 affine patches defined by:
x - y
Defn: Defined on coordinates by sending [y : u] to
[1 : y : u : 1]
Return True if self is presented as a subscheme of an ambient projective space.
OUTPUT:
Boolean.
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
Note that toric varieties are implemented differently than projective spaces. This is why this method returns False for toric varieties:
sage: PP.<x,y,z,w> = toric_varieties.P(3)
sage: V = PP.subscheme(x^3 + y^3 + z^3 + w^3)
sage: V.is_projective()
False
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
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.
Warning
You should not create objects of this class directly. The preferred method to construct such subschemes is to use complement() method of algebraic schemes.
OUTPUT:
An instance of AlgebraicScheme_quasi.
EXAMPLES:
sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: S = P.subscheme([])
sage: T = P.subscheme([x-y])
sage: T.complement(S)
Quasi-projective subscheme X - Y of Projective Space of dimension 2 over
Integer Ring, where X is defined by:
(no polynomials)
and Y is defined by:
x - y
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
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
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)]
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; 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:
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
Return the Jacobian ideal.
This is the ideal generated by
OUTPUT:
An ideal in the coordinate ring of the ambient space.
REFERENCES:
[LazarsfeldJacobian] | Robert Lazarsfeld: Positivity in algebraic geometry II; Positivity for Vector Bundles, and Multiplier Ideals, page 181. |
EXAMPLES:
sage: P3.<w,x,y,z> = ProjectiveSpace(3, QQ)
sage: twisted_cubic = P3.subscheme(matrix([[w, x, y],[x, y, z]]).minors(2))
sage: twisted_cubic.Jacobian()
Ideal (-x^2 + w*y, -x*y + w*z, -y^2 + x*z, x*z, -2*w*z, w*y, 3*w*y, -2*w*x,
w^2, y*z, -2*x*z, w*z, 3*w*z, -2*w*y, w*x, z^2, -2*y*z, x*z, 3*x*z, -2*w*z,
w*y) of Multivariate Polynomial Ring in w, x, y, z over Rational Field
sage: twisted_cubic.defining_ideal()
Ideal (-x^2 + w*y, -x*y + w*z, -y^2 + x*z) of Multivariate Polynomial Ring
in w, x, y, z over Rational Field
Return the matrix \(\frac{\partial f_i}{\partial x_j}\) of (formal) partial derivatives.
OUTPUT:
A matrix of polynomials.
EXAMPLES:
sage: P3.<w,x,y,z> = ProjectiveSpace(3, QQ)
sage: twisted_cubic = P3.subscheme(matrix([[w, x, y],[x, y, z]]).minors(2))
sage: twisted_cubic.Jacobian_matrix()
[ y -2*x w 0]
[ z -y -x w]
[ 0 z -2*y x]
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)!
Returns a new projective subscheme whose base ring is self coerced to R.
EXAMPLES:
sage: P.<x,y>=ProjectiveSpace(QQ,1)
sage: X=P.subscheme([3*x^2-y^2])
sage: H=Hom(X,X)
sage: X.change_ring(GF(3))
Closed subscheme of Projective Space of dimension 1 over Finite Field of size 3 defined by:
-y^2
Return the codimension of the algebraic subscheme.
OUTPUT:
Integer.
EXAMPLES:
sage: PP.<x,y,z,w,v> = ProjectiveSpace(4,QQ)
sage: V = PP.subscheme(x*y)
sage: V.codimension()
1
sage: V.dimension()
3
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
Return the ideal that defines this scheme as a subscheme of its ambient space.
OUTPUT:
An ideal in the coordinate ring of the 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
Return the polynomials that define this scheme as a subscheme of its ambient space.
OUTPUT:
A tuple of polynomials in the coordinate ring of the 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)
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
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()
[
]
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
]
Return the rational points on the algebraic subscheme.
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:
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
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
Bases: sage.schemes.generic.algebraic_scheme.AlgebraicScheme_subscheme
Construct an algebraic subscheme of affine space.
Warning
You should not create objects of this class directly. The preferred method to construct such subschemes is to use subscheme() method of affine space.
INPUT:
EXAMPLES:
sage: A3.<x, y, z> = AffineSpace(3, QQ)
sage: A3.subscheme([x^2-y*z])
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
x^2 - y*z
TESTS:
sage: from sage.schemes.generic.algebraic_scheme import AlgebraicScheme_subscheme_affine
sage: AlgebraicScheme_subscheme_affine(A3, [x^2-y*z])
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
x^2 - y*z
Return the dimension of the affine algebraic subscheme.
OUTPUT:
Integer.
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
Test whether the algebraic subscheme is smooth.
INPUT:
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: A2.<x,y> = AffineSpace(2,QQ)
sage: cuspidal_curve = A2.subscheme([y^2-x^3])
sage: cuspidal_curve
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
-x^3 + y^2
sage: smooth_point = cuspidal_curve.point([1,1])
sage: smooth_point in cuspidal_curve
True
sage: singular_point = cuspidal_curve.point([0,0])
sage: singular_point in cuspidal_curve
True
sage: cuspidal_curve.is_smooth(smooth_point)
True
sage: cuspidal_curve.is_smooth(singular_point)
False
sage: cuspidal_curve.is_smooth()
False
Returns a morphism from this affine scheme into an ambient projective space of the same dimension.
INPUT:
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
(x : y : z : 1)
Bases: sage.schemes.generic.algebraic_scheme.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:
OUTPUT:
A algebraic subscheme of an affine toric variety.
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
Return the dimension of self.
OUTPUT:
EXAMPLES:
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: X.<x,y> = toric_varieties.A2_Z2()
sage: X.is_smooth()
False
sage: Y = X.subscheme([x*y, x^2])
sage: Y
Closed subscheme of 2-d affine toric variety defined by:
x*y,
x^2
sage: Y.dimension()
1
Test whether the algebraic subscheme is smooth.
INPUT:
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: 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: 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
Bases: sage.schemes.generic.algebraic_scheme.AlgebraicScheme_subscheme
Construct an algebraic subscheme of projective space.
Warning
You should not create objects of this class directly. The preferred method to construct such subschemes is to use subscheme() method of projective space.
INPUT:
EXAMPLES:
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
TESTS:
sage: from sage.schemes.generic.algebraic_scheme import AlgebraicScheme_subscheme_projective
sage: AlgebraicScheme_subscheme_projective(P, [x^2-y*z])
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x^2 - y*z
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:
OUTPUT:
An affine algebraic scheme with fixed embedding_morphism() equal to the default 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.embedding_morphism()
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 : x0 : x1)
sage: U.projective_embedding() is U.embedding_morphism()
True
Return the dimension of the projective algebraic subscheme.
OUTPUT:
Integer.
EXAMPLES:
sage: P2.<x,y,z> = ProjectiveSpace(2, QQ)
sage: P2.subscheme([]).dimension()
2
sage: P2.subscheme([x]).dimension()
1
sage: P2.subscheme([x^5]).dimension()
1
sage: P2.subscheme([x^2 + y^2 - z^2]).dimension()
1
sage: P2.subscheme([x*(x-z), y*(y-z)]).dimension()
0
Something less obvious:
sage: P3.<x,y,z,w,t> = ProjectiveSpace(4, QQ)
sage: X = P3.subscheme([x^2, x^2*y^2 + z^2*t^2, z^2 - w^2, 10*x^2 + w^2 - z^2])
sage: X
Closed subscheme of Projective Space of dimension 4 over Rational Field defined by:
x^2,
x^2*y^2 + z^2*t^2,
z^2 - w^2,
10*x^2 - z^2 + w^2
sage: X.dimension()
1
Test whether the algebraic subscheme is smooth.
INPUT:
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: P2.<x,y,z> = ProjectiveSpace(2,QQ)
sage: cuspidal_curve = P2.subscheme([y^2*z-x^3])
sage: cuspidal_curve
Closed subscheme of Projective Space of dimension 2 over Rational Field 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
sage: P2.subscheme([y^2*z-x^3+z^3+1/10*x*y*z]).is_smooth()
True
TESTS:
sage: H = P2.subscheme(x)
sage: H.is_smooth() # one of the few cases where the cone over the subvariety is smooth
True
Return an affine algebraic subscheme isomorphic to a neighborhood of the point.
INPUT:
OUTPUT
An affine algebraic scheme (polynomial equations in affine space) result such that
EXAMPLES:
sage: P.<x,y,z>= ProjectiveSpace(QQ,2)
sage: S = P.subscheme(x+2*y+3*z)
sage: s = S.point([0,-3,2]); s
(0 : -3/2 : 1)
sage: patch = S.neighborhood(s); patch
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x0 + 3*x1
sage: patch.embedding_morphism()
Scheme morphism:
From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x0 + 3*x1
To: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x + 2*y + 3*z
Defn: Defined on coordinates by sending (x0, x1) to
(x0 : -3/2 : x1 + 1)
sage: patch.embedding_center()
(0, 0)
sage: patch.embedding_morphism()([0,0])
(0 : -3/2 : 1)
sage: patch.embedding_morphism()(patch.embedding_center())
(0 : -3/2 : 1)
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:
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
Return the affine patch corresponding to cone as an affine algebraic scheme.
INPUT:
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. http://www.arxiv.org/abs/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: 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)])
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
Return the i-th affine patch of self as an affine toric algebraic scheme.
INPUT:
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
Return the dimension of self.
OUTPUT:
Integer. If self is empty, \(-1\) is returned.
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()
1
sage: P1xP1.subscheme([z0-z2, (z0-z2)^2]).dimension()
1
sage: P1xP1.subscheme([z0,z2]).dimension()
-1
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
Test whether the algebraic subscheme is smooth.
INPUT:
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: 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()
True
A more complicated example:
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: lp = LatticePolytope(matrix([(1,0,0),(1,1,0),(1,1,1),(1,0,1),(-2,-1,-1)]).transpose())
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
Return an toric algebraic scheme isomorphic to neighborhood of the point.
INPUT:
OUTPUT
An affine toric algebraic scheme (polynomial equations in an affine toric variety) with fixed embedding_morphism() and embedding_center().
EXAMPLES:
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: 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]
Test whether x is an algebraic scheme.
INPUT:
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