Algebraic schemes#
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
), orToric 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 introduced.
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() # needs sage.libs.singular
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 # needs sage.modules
[-x1^2 + x0*x2, -x1*x2 + x0*x3, -x2^2 + x1*x3]
sage: twisted_cubic = P3.subscheme(Q); twisted_cubic # needs sage.libs.singular sage.modules
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() # needs sage.libs.singular sage.modules
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); patch # needs sage.libs.singular sage.modules
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
-x1^2 + x2,
-x1*x2 + x3,
-x2^2 + x1*x3
sage: patch.embedding_morphism() # needs sage.libs.singular sage.modules
Scheme morphism:
From: Closed subscheme of Affine Space of dimension 3 over Rational Field
defined by: -x1^2 + x2, -x1*x2 + x3, -x2^2 + x1*x3
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 (x1, x2, x3) to (1 : x1 : x2 : x3)
AUTHORS:
David Kohel, William Stein (2005): initial version
Andrey Novoseltsev (2010-05-17): subschemes of toric varieties
Volker Braun (2010-12-24): documentation of schemes and refactoring; added coordinate neighborhoods and is_smooth()
Ben Hutz (2014): subschemes of Cartesian products of projective space
Ben Hutz (2017): split subschemes types into respective folders
- class sage.schemes.generic.algebraic_scheme.AlgebraicScheme(A, category=None)#
Bases:
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.
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)
- embedding_center()#
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 byembedding_center()
.OUTPUT:
A point of
self
. This raisesAttributeError
if there is no distinguished point, depending on howself
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: w^2*y^2 - x^2*y^2 + 6*w^2*y - 6*x^2*y + 2*w*y^2 + 2*x*y^2 - 7*w^2 + 7*x^2 + 12*w*y + 12*x*y - 14*w - 14*x 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: w^2*y^2 - x^2*y^2 + 6*w^2*y - 6*x^2*y + 2*w*y^2 + 2*x*y^2 - 7*w^2 + 7*x^2 + 12*w*y + 12*x*y - 14*w - 14*x 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 (w, x, y) to (w + 1 : x - 1 : y + 3 : 4)
- embedding_morphism()#
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 byembedding_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()
isself
.By default, it is the tautological embedding into its own ambient space
ambient_space()
.If the algebraic scheme (which itself is a subscheme of an auxiliary
ambient_space()
) was constructed as a patch or neighborhood of a point then the embedding is the embedding into the original scheme.A
NotImplementedError
is raised if the construction of the embedding morphism is not implemented yet.
EXAMPLES:
sage: A2.<x,y> = AffineSpace(QQ, 2) sage: C = A2.subscheme(x^2 + y^2 - 1) sage: C.embedding_morphism() # needs sage.libs.singular 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: # needs sage.geometry.polyhedron sage.graphs sage.libs.singular 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); nbhd Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: -y^2*z - 2*y*z
Note that \(p=(1,1,0)\) is a singular point of \(X\). So the neighborhood of \(p\) is not just affine space. The
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: -y^2*z - 2*y*z 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 (y, z) to (1 : y + 1 : z)
A couple more examples:
sage: # needs sage.geometry.polyhedron sage.graphs sage.libs.singular sage: patch1 = P1xP1.affine_patch(1); 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); 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]
- identity_morphism()#
Return the identity morphism.
OUTPUT: the identity morphism of the scheme
self
EXAMPLES:
sage: X = Spec(QQ) sage: X.identity_morphism() Scheme endomorphism of Spectrum of Rational Field Defn: Identity map
- is_projective()#
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: # needs sage.geometry.polyhedron sage.graphs 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
- 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:
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
- 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(**kwds)#
Return the set of rational points on this algebraic scheme over the field \(F\).
INPUT:
kwds:
bound
- integer (optional, default=0). The bound for the coordinates for subschemes with dimension at least 1.F
- field (optional, default=base ring). The field to compute the rational points over.
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(F=GF(7^2, 'b')) # needs sage.rings.finite_rings [(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, category=None)#
Bases:
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
- Jacobian()#
Return the Jacobian ideal.
This is the ideal generated by
the \(d\times d\) minors of the Jacobian matrix, where \(d\) is the
codimension()
of the algebraic scheme, andthe defining polynomials of the algebraic scheme. Note that some authors do not include these in the definition of the Jacobian ideal. An example of a reference that does include the defining equations is [Laz2004], p. 181.
OUTPUT:
An ideal in the coordinate ring of the ambient space.
EXAMPLES:
sage: P3.<w,x,y,z> = ProjectiveSpace(3, QQ) sage: twisted_cubic = P3.subscheme(matrix([[w, x, y], # needs sage.libs.singular ....: [x, y, z]]).minors(2)) sage: twisted_cubic.Jacobian() # needs sage.libs.singular 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() # needs sage.libs.singular 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
This example addresses issue github issue #20512:
sage: X = P3.subscheme([]) sage: X.Jacobian() == P3.coordinate_ring().unit_ideal() # needs sage.libs.singular True
- Jacobian_matrix()#
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], # needs sage.libs.singular ....: [x, y, z]]).minors(2)) sage: twisted_cubic.Jacobian_matrix() # needs sage.libs.singular [ y -2*x w 0] [ z -y -x w] [ 0 z -2*y x]
This example addresses issue github issue #20512:
sage: X = P3.subscheme([]) sage: X.Jacobian_matrix().base_ring() == P3.coordinate_ring() # needs sage.libs.singular True
- 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')) # needs sage.rings.finite_rings 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)!
- change_ring(R)#
Returns a new algebraic subscheme which is this subscheme coerced to
R
.INPUT:
R
– ring or morphism.
OUTPUT:
A new algebraic subscheme which is this subscheme 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
sage: # needs sage.rings.number_field sage: K.<w> = QuadraticField(2) sage: R.<z> = K[] sage: L.<v> = K.extension(z^3 - 5) sage: P.<x,y> = ProjectiveSpace(K, 1) sage: X = P.subscheme(x - w*y) # needs sage.libs.singular sage: X.change_ring(L) # needs sage.libs.singular Closed subscheme of Projective Space of dimension 1 over Number Field in v with defining polynomial z^3 - 5 over its base field defined by: x + (-w)*y
sage: # needs sage.rings.number_field sage: K.<w> = QuadraticField(2) sage: R.<z> = K[] sage: L.<v> = K.extension(z^3 - 5) sage: P.<x,y,z> = AffineSpace(L, 3) sage: X = P.subscheme([x - w*y, z^2 - v*x]) # needs sage.libs.singular sage: emb = L.embeddings(QQbar) # needs sage.libs.singular sage: X.change_ring(emb[0]) # needs sage.libs.singular Closed subscheme of Affine Space of dimension 3 over Algebraic Field defined by: x + (-1.414213562373095? + 0.?e-16*I)*y, z^2 + (0.8549879733383485? + 1.480882609682365?*I)*x
sage: # needs sage.rings.number_field sage: K.<w> = QuadraticField(2) sage: R.<z> = K[] sage: L.<v> = K.extension(z^3 - 5) sage: P.<x,y,z> = AffineSpace(L, 3) sage: X = P.subscheme([x - w*y, z^2 - v*x]) # needs sage.libs.singular sage: emb = L.embeddings(QQbar) # needs sage.libs.singular sage: X.change_ring(emb[1]) # needs sage.libs.singular Closed subscheme of Affine Space of dimension 3 over Algebraic Field defined by: x + (-1.414213562373095? + 0.?e-16*I)*y, z^2 + (0.8549879733383485? - 1.480882609682365?*I)*x
sage: # needs sage.rings.number_field sage: K.<w> = QuadraticField(-3) sage: P.<x,y> = ProjectiveSpace(K, 1) sage: X = P.subscheme(x - w*y) # needs sage.libs.singular sage: X.change_ring(CC) # needs sage.libs.singular Closed subscheme of Projective Space of dimension 1 over Complex Field with 53 bits of precision defined by: x + (-1.73205080756888*I)*y
sage: # needs sage.rings.number_field sage: K.<w> = QuadraticField(3) sage: P.<x,y> = ProjectiveSpace(K, 1) sage: X = P.subscheme(x - w*y) # needs sage.libs.singular sage: X.change_ring(RR) # needs sage.libs.singular Closed subscheme of Projective Space of dimension 1 over Real Field with 53 bits of precision defined by: x - 1.73205080756888*y
sage: # needs sage.rings.number_field sage: K.<v> = CyclotomicField(7) sage: O = K.maximal_order() sage: P.<x,y> = ProjectiveSpace(O, 1) sage: X = P.subscheme([x^2 + O(v)*y^2]) # needs sage.libs.singular sage: X.change_ring(CC) # needs sage.libs.singular Closed subscheme of Projective Space of dimension 1 over Complex Field with 53 bits of precision defined by: x^2 + (0.623489801858734 + 0.781831482468030*I)*y^2 sage: X.change_ring(K).change_ring(K.embeddings(QQbar)[3]) # needs sage.libs.singular Closed subscheme of Projective Space of dimension 1 over Algebraic Field defined by: x^2 + (-0.9009688679024191? - 0.4338837391175581?*I)*y^2
sage: # needs sage.rings.number_field sage: R.<x> = QQ[] sage: f = x^6 - 2 sage: L.<b> = NumberField(f, embedding=f.roots(CC)[2][0]) sage: A.<x,y> = AffineSpace(L, 2) sage: H = Hom(A, A) sage: X = A.subscheme([b*x^2, y^2]) # needs sage.libs.singular sage: X.change_ring(CC) # needs sage.libs.singular Closed subscheme of Affine Space of dimension 2 over Complex Field with 53 bits of precision defined by: (-0.561231024154687 - 0.972080648619833*I)*x^2, y^2
- codimension()#
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() # needs sage.libs.singular 1 sage: V.dimension() # needs sage.libs.singular 3
- 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.
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
- defining_polynomials()#
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)
- 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() # needs sage.libs.singular [ 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 is not accidentally returned (see github issue #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() # needs sage.libs.singular [ ]
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() # needs sage.libs.singular [ Closed subscheme of Affine Space of dimension 4 over Rational Field defined by: w, z, y, x ]
- is_irreducible()#
Return whether this subscheme is or is not irreducible.
OUTPUT: Boolean.
EXAMPLES:
sage: # needs sage.rings.number_field sage: K = QuadraticField(-3) sage: P.<x,y,z,w,t,u> = ProjectiveSpace(K, 5) sage: X = P.subscheme([x*y - z^2 - K.0*t^2, t*w*x + y*z^2 - u^3]) sage: X.is_irreducible() # needs sage.libs.singular True
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: X = P.subscheme([(y + x - z)^2]) sage: X.is_irreducible() # needs sage.libs.singular False
sage: A.<x,y,z,w> = AffineSpace(GF(17), 4) sage: X = A.subscheme([ ....: x*y*z^2 - x*y*z*w - z*w^2 + w^3, ....: x^3*y*z*w - x*y^3*z - x^2*y*z*w - x^2*w^3 + y^2*w^2 + x*w^3 ....: ]) sage: X.is_irreducible() # needs sage.libs.singular False
- normalize_defining_polynomials()#
Function to normalize the coefficients of defining polynomials of given subscheme.
Normalization as in removing denominator from all the coefficients, and then removing any common factor between the coefficients. It takes LCM of denominators and then removes common factor among coefficients, if any.
EXAMPLES:
sage: A.<x,y> = AffineSpace(2, QQ) sage: S = A.subscheme([2*x^2 + 4*x*y, 1/8*x + 1/3*y]) sage: S.normalize_defining_polynomials() sage: S.defining_polynomials() (x^2 + 2*x*y, 3*x + 8*y)
- rational_points(**kwds)#
Return the rational points on the algebraic subscheme.
For a dimension 0 subscheme, if the base ring is a numerical field such as the ComplexField the results returned could be very far from correct. If the polynomials defining the subscheme are defined over a number field, you will get better results calling rational points with \(F\) defined as the number field and the base ring as the field of definition. If the base ring is a number field, the embedding into
F
must be known.In the case of numerically approximated points, the points are returned over as points of the ambient space.
For a dimension greater than 0 scheme, depending on bound size, either the points in the ambient space are enumerated or a sieving algorithm lifting points modulo primes is used. See the documentation in homset for the details of the sieving algorithm.
INPUT:
kwds:
bound
- integer (optional, default=0). The bound for the coordinates for subschemes with dimension at least 1.prec
- integer (optional, default=53). The precision to use to compute the elements of bounded height for number fields.F
- field (optional, default=base ring). The field to compute the rational points over.point_tolerance
- positive real number (optional, default=10^(-10)). For numerically inexact fields, two points are considered the same if their coordinates are within tolerance.zero_tolerance
- positive real number (optional, default=10^(-10)). For numerically inexact fields, points are on the subscheme if they satisfy the equations to within tolerance.tolerance
- a rational number in (0,1] used in doyle-krumm algorithm-4
OUTPUT: list of points in subscheme or ambient space
Warning
For numerically inexact fields such as ComplexField or RealField the list of points returned is very likely to be incomplete at best.
EXAMPLES:
Enumerate over a projective scheme over a number field:
sage: # needs sage.rings.number_field sage: u = QQ['u'].0 sage: K.<v> = NumberField(u^2 + 3) sage: A.<x,y> = ProjectiveSpace(K, 1) sage: X = A.subscheme(x^2 - y^2) sage: X.rational_points(bound=3) # needs sage.libs.singular [(-1 : 1), (1 : 1)]
One can enumerate points up to a given bound on a projective scheme over the rationals:
sage: E = EllipticCurve('37a') # needs sage.schemes sage: E.rational_points(bound=8) # needs sage.schemes [(-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)) # needs sage.schemes sage: Etilde.rational_points() # needs sage.schemes [(0 : 1 : 0), (0 : 0 : 1), (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) # needs sage.schemes sage: C.rational_points() # needs sage.schemes [(0 : 1 : 0), (0 : 1 : 1), (0 : 6 : 1), (2 : 0 : 1), (4 : 0 : 1), (6 : 1 : 1), (6 : 6 : 1)]
sage: # needs sage.rings.number_field sage: K.<v> = QuadraticField(-3) sage: P.<x,y,z> = ProjectiveSpace(K, 2) sage: X = P.subscheme([x^2 - v^2*x*z, y*x - v*z^2]) sage: X.rational_points(F=CC) [(-3.00000000000000 : -0.577350269189626*I : 1.00000000000000), (0.000000000000000 : 1.00000000000000 : 0.000000000000000)]
sage: # needs sage.rings.number_field sage: K.<v> = QuadraticField(3) sage: A.<x,y> = AffineSpace(K, 2) sage: X = A.subscheme([x^2 - v^2*y, y*x - v]) sage: X.rational_points(F=RR) [(1.73205080756888, 1.00000000000000)]
Todo
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() # needs sage.libs.singular 1
Then we compute the corresponding reduced scheme:
sage: Y = X.reduce(); Y # needs sage.libs.singular 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: # needs sage.libs.singular 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
- specialization(D=None, phi=None)#
Specialization of this subscheme.
Given a family of maps defined over a polynomial ring. A specialization is a particular member of that family. The specialization can be specified either by a dictionary or a
SpecializationMorphism
.INPUT:
D
– dictionary (optional)phi
–SpecializationMorphism
(optional)
OUTPUT:
SchemeMorphism_polynomial
EXAMPLES:
sage: R.<c> = PolynomialRing(QQ) sage: P.<x,y> = ProjectiveSpace(R, 1) sage: X = P.subscheme([x^2 + c*y^2]) sage: X.specialization(dict({c:2})) Closed subscheme of Projective Space of dimension 1 over Rational Field defined by: x^2 + 2*y^2
sage: R.<c> = PolynomialRing(QQ) sage: S.<a,b> = R[] sage: P.<x,y,z> = AffineSpace(S, 3) sage: X = P.subscheme([x^2 + a*c*y^2 - b*z^2]) sage: from sage.rings.polynomial.flatten import SpecializationMorphism sage: phi = SpecializationMorphism(P.coordinate_ring(), ....: dict({c: 2, a: 1})) sage: X.specialization(phi=phi) # needs sage.libs.singular Closed subscheme of Affine Space of dimension 3 over Univariate Polynomial Ring in b over Rational Field defined by: x^2 + 2*y^2 + (-b)*z^2
- 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 # needs sage.libs.singular 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() # needs sage.libs.singular 1 sage: S.reduce() # needs sage.libs.singular 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)]) # needs sage.libs.singular 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 # needs sage.libs.singular True
- weil_restriction()#
Compute the Weil restriction of this variety over some extension field. If the field is a finite field, then this computes the Weil restriction to the prime subfield.
A Weil restriction of scalars - denoted \(Res_{L/k}\) - is a functor which, for any finite extension of fields \(L/k\) and any algebraic variety \(X\) over \(L\), produces another corresponding variety \(Res_{L/k}(X)\), defined over \(k\). It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields.
This function does not compute this Weil restriction directly but computes on generating sets of polynomial ideals:
Let \(d\) be the degree of the field extension \(L/k\), let \(a\) a generator of \(L/k\) and \(p\) the minimal polynomial of \(L/k\). Denote this ideal by \(I\).
Specifically, this function first maps each variable \(x\) to its representation over \(k\): \(\sum_{i=0}^{d-1} a^i x_i\). Then each generator of \(I\) is evaluated over these representations and reduced modulo the minimal polynomial \(p\). The result is interpreted as a univariate polynomial in \(a\) and its coefficients are the new generators of the returned ideal.
If the input and the output ideals are radical, this is equivalent to the statement about algebraic varieties above.
OUTPUT: Affine subscheme - the Weil restriction of
self
.EXAMPLES:
sage: # needs sage.rings.number_field sage: R.<x> = QQ[] sage: K.<w> = NumberField(x^5 - 2) sage: R.<x> = K[] sage: L.<v> = K.extension(x^2 + 1) sage: A.<x,y> = AffineSpace(L, 2) sage: X = A.subscheme([y^2 - L(w)*x^3 - v]) # needs sage.libs.singular sage: X.weil_restriction() # needs sage.libs.singular Closed subscheme of Affine Space of dimension 4 over Number Field in w with defining polynomial x^5 - 2 defined by: (-w)*z0^3 + (3*w)*z0*z1^2 + z2^2 - z3^2, (-3*w)*z0^2*z1 + w*z1^3 + 2*z2*z3 - 1 sage: X.weil_restriction().ambient_space() is A.weil_restriction() # needs sage.libs.singular True
sage: A.<x,y,z> = AffineSpace(GF(5^2, 't'), 3) # needs sage.rings.finite_rings sage: X = A.subscheme([y^2 - x*z, z^2 + 2*y]) # needs sage.libs.singular sage.rings.finite_rings sage: X.weil_restriction() # needs sage.libs.singular sage.rings.finite_rings Closed subscheme of Affine Space of dimension 6 over Finite Field of size 5 defined by: z2^2 - 2*z3^2 - z0*z4 + 2*z1*z5, 2*z2*z3 + z3^2 - z1*z4 - z0*z5 - z1*z5, z4^2 - 2*z5^2 + 2*z2, 2*z4*z5 + z5^2 + 2*z3
- sage.schemes.generic.algebraic_scheme.is_AlgebraicScheme(x)#
Test whether
x
is an algebraic scheme.INPUT:
x
– anything.
OUTPUT:
Boolean. Whether
x
is 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