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# 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 \(S\), or anything that can be converted into such an ideal

  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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