Algebraic schemes¶

An algebraic scheme must be defined by sets of equations in affine or projective spaces, perhaps by means of gluing relations.

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.

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)
...
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)
[(0 : 0 : 1),
 (1 : 0 : 1),
 (-1 : 0 : 1),
 (0 : -1 : 1),
 (1 : -1 : 1),
 (-1 : -1 : 1),
 (2 : 2 : 1),
 (2 : -3 : 1),
 (1/4 : -3/8 : 1),
 (1/4 : -5/8 : 1),
 (0 : 1 : 0)]

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()
[(2 : 0 : 1), (4 : 0 : 1), (0 : 1 : 1), (6 : 1 : 1), (0 : 6 : 1), (6 : 6 : 1), (0 : 1 : 0)]

TODO:

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.
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

sage.schemes.generic.algebraic_scheme.is_AlgebraicScheme(x)¶

Return True if $x$ is an algebraic scheme, i.e., a subscheme of an ambient space over a ring defined by polynomial equations.

EXAMPLES: 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

Algebraic schemes¶

Previous topic

Next topic

This Page

Navigation

Algebraic schemes¶

Previous topic

Next topic

This Page

Quick search

Navigation