Base class for points on a scheme, either topological or defined by a morphism.
INPUT:
TESTS:
sage: from sage.schemes.generic.point import SchemePoint
sage: S = Spec(ZZ)
sage: P = SchemePoint(S); P
Point on Spectrum of Integer Ring
Return a string representation of this generic scheme point.
TESTS:
sage: from sage.schemes.generic.point import SchemePoint
sage: S = Spec(ZZ)
sage: P = SchemePoint(S); P
Point on Spectrum of Integer Ring
sage: P._repr_()
'Point on Spectrum of Integer Ring'
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
INPUT:
INPUT:
INPUT:
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)
...
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 Principal ideal (-x^2 + y*z) of Multivariate Polynomial Ring in x, y, z over Rational Field
Return a string representation of this scheme point.
TESTS:
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); pt
Point on Projective Space of dimension 2 over Rational Field defined by the Principal ideal (-x^2 + y*z) of Multivariate Polynomial Ring in x, y, z over Rational Field
sage: pt._repr_()
'Point on Projective Space of dimension 2 over Rational Field defined by the Principal ideal (-x^2 + y*z) of Multivariate Polynomial Ring in x, y, z over Rational Field'
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()
Principal ideal (-x^2 + y*z) of Multivariate Polynomial Ring in x, y, z over Rational Field