# Points on schemes¶

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

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)

INPUT:

• f - a morphism of schemes
morphism()
class sage.schemes.generic.point.SchemeTopologicalPoint(S)

Base class for topological points on schemes.

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

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)

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