Points on schemes - Schemes

Skip to content
Back to top
View this page
Edit this page

# Points on schemes

class sage.schemes.generic.point.SchemePoint(S, parent=None)

  Bases: `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

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

    Python

        >>> 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)

  Bases: `SchemePoint`

  Base class for topological points on schemes.

class sage.schemes.generic.point.SchemeTopologicalPoint_affine_open(u, x)

  Bases: `SchemeTopologicalPoint`

  INPUT:

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

  - `x` – topological 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: `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()

    Return the prime ideal that defines this scheme point.

    EXAMPLES:

    Sage

        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

    Python

        >>> 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