Points on schemes#

class sage.schemes.generic.point.SchemePoint(S, parent=None)[source]#

Bases: Element

Base class for points on a scheme, either topological or defined by a morphism.

scheme()[source]#

Return the scheme on which self is a point.

EXAMPLES:

sage: from sage.schemes.generic.point import SchemePoint
sage: S = Spec(ZZ)
sage: P = SchemePoint(S)
sage: P.scheme()
Spectrum of Integer Ring

>>> from sage.all import *
>>> from sage.schemes.generic.point import SchemePoint
>>> S = Spec(ZZ)
>>> P = SchemePoint(S)
>>> P.scheme()
Spectrum of Integer Ring
class sage.schemes.generic.point.SchemeTopologicalPoint(S)[source]#

Bases: SchemePoint

Base class for topological points on schemes.

class sage.schemes.generic.point.SchemeTopologicalPoint_affine_open(u, x)[source]#

Bases: SchemeTopologicalPoint

INPUT:

  • u – morphism with domain an affine scheme \(U\)

  • x – topological point on \(U\)

affine_open()[source]#

Return the affine open subset \(U\).

embedding_of_affine_open()[source]#

Return the embedding from the affine open subset \(U\) into this scheme.

point_on_affine()[source]#

Return the scheme point on the affine open \(U\).

class sage.schemes.generic.point.SchemeTopologicalPoint_prime_ideal(S, P, check=False)[source]#

Bases: SchemeTopologicalPoint

INPUT:

  • S – an affine scheme

  • P – a prime ideal of the coordinate ring of \(S\), or anything that can be converted into such an ideal

prime_ideal()[source]#

Return the prime ideal that defines this scheme point.

EXAMPLES:

sage: from sage.schemes.generic.point import SchemeTopologicalPoint_prime_ideal
sage: P2.<x, y, z> = ProjectiveSpace(2, QQ)
sage: pt = SchemeTopologicalPoint_prime_ideal(P2, y*z - x^2)
sage: pt.prime_ideal()
Ideal (-x^2 + y*z) of Multivariate Polynomial Ring in x, y, z over Rational Field

>>> from sage.all import *
>>> from sage.schemes.generic.point import SchemeTopologicalPoint_prime_ideal
>>> P2 = ProjectiveSpace(Integer(2), QQ, names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> pt = SchemeTopologicalPoint_prime_ideal(P2, y*z - x**Integer(2))
>>> pt.prime_ideal()
Ideal (-x^2 + y*z) of Multivariate Polynomial Ring in x, y, z over Rational Field
sage.schemes.generic.point.is_SchemeTopologicalPoint(x)[source]#