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 \(U\)
- class sage.schemes.generic.point.SchemeTopologicalPoint_prime_ideal(S, P, check=False)[source]#
Bases:
SchemeTopologicalPoint
INPUT:
S
– an affine schemeP
– 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