Points on schemes

class sage.schemes.generic.point.SchemePoint(S)

Bases: sage.structure.element.Element

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

scheme()

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
class sage.schemes.generic.point.SchemeRationalPoint(f)

Bases: sage.schemes.generic.point.SchemePoint

INPUT:

  • f - a morphism of schemes
morphism()

x.__init__(...) initializes x; see help(type(x)) for signature

class sage.schemes.generic.point.SchemeTopologicalPoint(S)

Bases: sage.schemes.generic.point.SchemePoint

INPUT:

  • S - a scheme

TESTS:

sage: from sage.schemes.generic.point import SchemePoint
sage: S = Spec(ZZ)
sage: P = SchemePoint(S); P
Point on Spectrum of Integer Ring
class sage.schemes.generic.point.SchemeTopologicalPoint_affine_open(u, x)

Bases: sage.schemes.generic.point.SchemeTopologicalPoint

INPUT:

  • u - morphism with domain U an affine scheme
  • x - point on U
affine_open()

Return the affine open subset U.

embedding_of_affine_open()

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

point_on_affine()

Return the scheme point on the affine open U.

class sage.schemes.generic.point.SchemeTopologicalPoint_prime_ideal(S, P, check=False)

Bases: sage.schemes.generic.point.SchemeTopologicalPoint

INPUT:

  • S - an affine scheme
  • P - a prime ideal of the coordinate ring of S

TESTS:

sage: from sage.schemes.generic.point import SchemeTopologicalPoint_prime_ideal
sage: S = Spec(ZZ)
sage: P = SchemeTopologicalPoint_prime_ideal(S, 3); P
Point on Spectrum of Integer Ring defined by the Principal ideal (3) of Integer Ring
sage: SchemeTopologicalPoint_prime_ideal(S, 6, check=True)
Traceback (most recent call last):
...
ValueError: The argument Principal ideal (6) of Integer Ring must be a prime ideal of Integer Ring
sage: SchemeTopologicalPoint_prime_ideal(S, ZZ.ideal(7))
Point on Spectrum of Integer Ring defined by the Principal ideal (7) of Integer Ring

We define a parabola in the projective plane as a point corresponding to a prime ideal:

sage: P2.<x, y, z> = ProjectiveSpace(2, QQ)
sage: SchemeTopologicalPoint_prime_ideal(P2, y*z-x^2)
Point on Projective Space of dimension 2 over Rational Field defined by the Ideal (-x^2 + y*z) of Multivariate Polynomial Ring in x, y, z over Rational Field
prime_ideal()

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
sage.schemes.generic.point.is_SchemeRationalPoint(x)

x.__init__(...) initializes x; see help(type(x)) for signature

sage.schemes.generic.point.is_SchemeTopologicalPoint(x)

x.__init__(...) initializes x; see help(type(x)) for signature

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