Set of homomorphisms between two projective schemes

For schemes \(X\) and \(Y\), this module implements the set of morphisms \(Hom(X,Y)\). This is done by SchemeHomset_generic.

As a special case, the Hom-sets can also represent the points of a scheme. Recall that the \(K\)-rational points of a scheme \(X\) over \(k\) can be identified with the set of morphisms \(Spec(K) \to X\). In Sage the rational points are implemented by such scheme morphisms. This is done by SchemeHomset_points and its subclasses.

Note

You should not create the Hom-sets manually. Instead, use the Hom() method that is inherited by all schemes.

AUTHORS:

• William Stein (2006): initial version.
• Volker Braun (2011-08-11): significant improvement and refactoring.
• Ben Hutz (June 2012): added support for projective ring

class sage.schemes.projective.projective_homset.SchemeHomset_points_abelian_variety_field(X, Y, category=None, check=True, base=Integer Ring)

Bases: sage.schemes.projective.projective_homset.SchemeHomset_points_projective_field

Set of rational points of an abelian variety.

INPUT:

See SchemeHomset_generic.

TESTS:

The bug reported at trac #1785 is fixed:

sage: K.<a> = NumberField(x^2 + x - (3^3-3))
sage: E = EllipticCurve('37a')
sage: X = E(K)
sage: X
Abelian group of points on Elliptic Curve defined by
y^2 + y = x^3 + (-1)*x over Number Field in a with
defining polynomial x^2 + x - 24
sage: P = X([3,a])
sage: P
(3 : a : 1)
sage: P in E
False
sage: P in E.base_extend(K)
True
sage: P in X.codomain()
False
sage: P in X.extended_codomain()
True

base_extend(R)

Extend the base ring.

This is currently not implemented except for the trivial case R==ZZ.

INPUT:

• R – a ring.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: Hom = E.point_homset();  Hom
Abelian group of points on Elliptic Curve defined
by y^2 + y = x^3 - x over Rational Field
sage: Hom.base_ring()
Integer Ring
sage: Hom.base_extend(QQ)
Traceback (most recent call last):
...
NotImplementedError: Abelian variety point sets are not
implemented as modules over rings other than ZZ.

class sage.schemes.projective.projective_homset.SchemeHomset_points_projective_field(X, Y, category=None, check=True, base=Integer Ring)

Bases: sage.schemes.generic.homset.SchemeHomset_points

Set of rational points of a projective variety over a field.

INPUT:

See SchemeHomset_generic.

EXAMPLES:

sage: from sage.schemes.projective.projective_homset import SchemeHomset_points_projective_field
sage: SchemeHomset_points_projective_field(Spec(QQ), ProjectiveSpace(QQ,2))
Set of rational points of Projective Space of dimension 2 over Rational Field

points(B=0)

Return some or all rational points of a projective scheme.

INPUT:

• \(B\) – integer (optional, default=0). The bound for the coordinates.

OUTPUT:

A list of points. Over a finite field, all points are returned. Over an infinite field, all points satisfying the bound are returned.

EXAMPLES:

sage: P1 = ProjectiveSpace(GF(2),1)
sage: F.<a> = GF(4,'a')
sage: P1(F).points()
[(0 : 1), (1 : 0), (1 : 1), (a : 1), (a + 1 : 1)]

class sage.schemes.projective.projective_homset.SchemeHomset_points_projective_ring(X, Y, category=None, check=True, base=Integer Ring)

Bases: sage.schemes.generic.homset.SchemeHomset_points

Set of rational points of a projective variety over a commutative ring.

INPUT:

See SchemeHomset_generic.

EXAMPLES:

sage: from sage.schemes.projective.projective_homset import SchemeHomset_points_projective_ring
sage: SchemeHomset_points_projective_ring(Spec(ZZ), ProjectiveSpace(ZZ,2))
Set of rational points of Projective Space of dimension 2 over Integer Ring

points(B=0)

Return some or all rational points of a projective scheme.

INPUT:

• \(B\) – integer (optional, default=0). The bound for the coordinates.

EXAMPLES:

sage: from sage.schemes.projective.projective_homset import SchemeHomset_points_projective_ring
sage: H = SchemeHomset_points_projective_ring(Spec(ZZ), ProjectiveSpace(ZZ,2))
sage: H.points(3)
[(0 : 0 : 1), (0 : 1 : -3), (0 : 1 : -2), (0 : 1 : -1), (0 : 1 : 0), (0
: 1 : 1), (0 : 1 : 2), (0 : 1 : 3), (0 : 2 : -3), (0 : 2 : -1), (0 : 2 :
1), (0 : 2 : 3), (0 : 3 : -2), (0 : 3 : -1), (0 : 3 : 1), (0 : 3 : 2),
(1 : -3 : -3), (1 : -3 : -2), (1 : -3 : -1), (1 : -3 : 0), (1 : -3 : 1),
(1 : -3 : 2), (1 : -3 : 3), (1 : -2 : -3), (1 : -2 : -2), (1 : -2 : -1),
(1 : -2 : 0), (1 : -2 : 1), (1 : -2 : 2), (1 : -2 : 3), (1 : -1 : -3),
(1 : -1 : -2), (1 : -1 : -1), (1 : -1 : 0), (1 : -1 : 1), (1 : -1 : 2),
(1 : -1 : 3), (1 : 0 : -3), (1 : 0 : -2), (1 : 0 : -1), (1 : 0 : 0), (1
: 0 : 1), (1 : 0 : 2), (1 : 0 : 3), (1 : 1 : -3), (1 : 1 : -2), (1 : 1 :
-1), (1 : 1 : 0), (1 : 1 : 1), (1 : 1 : 2), (1 : 1 : 3), (1 : 2 : -3),
(1 : 2 : -2), (1 : 2 : -1), (1 : 2 : 0), (1 : 2 : 1), (1 : 2 : 2), (1 :
2 : 3), (1 : 3 : -3), (1 : 3 : -2), (1 : 3 : -1), (1 : 3 : 0), (1 : 3 :
1), (1 : 3 : 2), (1 : 3 : 3), (2 : -3 : -3), (2 : -3 : -2), (2 : -3 :
-1), (2 : -3 : 0), (2 : -3 : 1), (2 : -3 : 2), (2 : -3 : 3), (2 : -2 :
-3), (2 : -2 : -1), (2 : -2 : 1), (2 : -2 : 3), (2 : -1 : -3), (2 : -1 :
-2), (2 : -1 : -1), (2 : -1 : 0), (2 : -1 : 1), (2 : -1 : 2), (2 : -1 :
3), (2 : 0 : -3), (2 : 0 : -1), (2 : 0 : 1), (2 : 0 : 3), (2 : 1 : -3),
(2 : 1 : -2), (2 : 1 : -1), (2 : 1 : 0), (2 : 1 : 1), (2 : 1 : 2), (2 :
1 : 3), (2 : 2 : -3), (2 : 2 : -1), (2 : 2 : 1), (2 : 2 : 3), (2 : 3 :
-3), (2 : 3 : -2), (2 : 3 : -1), (2 : 3 : 0), (2 : 3 : 1), (2 : 3 : 2),
(2 : 3 : 3), (3 : -3 : -2), (3 : -3 : -1), (3 : -3 : 1), (3 : -3 : 2),
(3 : -2 : -3), (3 : -2 : -2), (3 : -2 : -1), (3 : -2 : 0), (3 : -2 : 1),
(3 : -2 : 2), (3 : -2 : 3), (3 : -1 : -3), (3 : -1 : -2), (3 : -1 : -1),
(3 : -1 : 0), (3 : -1 : 1), (3 : -1 : 2), (3 : -1 : 3), (3 : 0 : -2), (3
: 0 : -1), (3 : 0 : 1), (3 : 0 : 2), (3 : 1 : -3), (3 : 1 : -2), (3 : 1
: -1), (3 : 1 : 0), (3 : 1 : 1), (3 : 1 : 2), (3 : 1 : 3), (3 : 2 : -3),
(3 : 2 : -2), (3 : 2 : -1), (3 : 2 : 0), (3 : 2 : 1), (3 : 2 : 2), (3 :
2 : 3), (3 : 3 : -2), (3 : 3 : -1), (3 : 3 : 1), (3 : 3 : 2)]