Closed points of integral curves

A rational point of a curve in Sage is represented by its coordinates. If the curve is defined over finite field and integral, that is reduced and irreducible, then it is empowered by the global function field machinery of Sage. Thus closed points of the curve are computable, as represented by maximal ideals of the coordinate ring of the ambient space.

EXAMPLES:

sage: F.<a> = GF(2)
sage: P.<x,y> = AffineSpace(F, 2);
sage: C = Curve(y^2 + y - x^3)
sage: C.closed_points()
[Point (x, y), Point (x, y + 1)]
sage: C.closed_points(2)
[Point (y^2 + y + 1, x + 1),
 Point (y^2 + y + 1, x + y),
 Point (y^2 + y + 1, x + y + 1)]
sage: C.closed_points(3)
[Point (x^2 + x + y, x*y + 1, y^2 + x + 1),
 Point (x^2 + x + y + 1, x*y + x + 1, y^2 + x)]

Closed points of projective curves are represented by homogeneous maximal ideals:

sage: F.<a> = GF(2)
sage: P.<x,y,z> = ProjectiveSpace(F, 2)
sage: C = Curve(x^3*y + y^3*z + x*z^3)
sage: C.closed_points()
[Point (x, z), Point (x, y), Point (y, z)]
sage: C.closed_points(2)
[Point (y^2 + y*z + z^2, x + y + z)]
sage: C.closed_points(3)
[Point (y^3 + y^2*z + z^3, x + y),
 Point (y^3 + y*z^2 + z^3, x + z),
 Point (x^2 + x*z + y*z + z^2, x*y + x*z + z^2, y^2 + x*z),
 Point (x^2 + y*z, x*y + x*z + z^2, y^2 + x*z + y*z),
 Point (x^3 + x*z^2 + z^3, y + z),
 Point (x^2 + y*z + z^2, x*y + x*z + y*z, y^2 + x*z + y*z + z^2),
 Point (x^2 + y*z + z^2, x*y + z^2, y^2 + x*z + y*z)]

Rational points are easily converted to closed points and vice versa if the closed point is of degree one:

sage: F.<a> = GF(2)
sage: P.<x,y,z> = ProjectiveSpace(F, 2)
sage: C = Curve(x^3*y + y^3*z + x*z^3)
sage: p1, p2, p3 = C.closed_points()
sage: p1.rational_point()
(0 : 1 : 0)
sage: p2.rational_point()
(0 : 0 : 1)
sage: p3.rational_point()
(1 : 0 : 0)
sage: _.closed_point()
Point (y, z)
sage: _ == p3
True

AUTHORS:

  • Kwankyu Lee (2019-03): initial version

class sage.schemes.curves.closed_point.CurveClosedPoint(S, P, check=False)

Bases: sage.schemes.generic.point.SchemeTopologicalPoint_prime_ideal

Base class of closed points of curves.

class sage.schemes.curves.closed_point.IntegralAffineCurveClosedPoint(curve, prime_ideal, degree)

Bases: sage.schemes.curves.closed_point.IntegralCurveClosedPoint

Closed points of affine curves.

projective(i=0)

Return the point in the projective closure of the curve, of which this curve is the i-th affine patch.

INPUT:

  • i – an integer

EXAMPLES:

sage: F.<a> = GF(2)
sage: A.<x,y> = AffineSpace(F, 2)
sage: C = Curve(y^2 + y - x^3, A)
sage: p1, p2 = C.closed_points()
sage: p1
Point (x, y)
sage: p2
Point (x, y + 1)
sage: p1.projective()
Point (x1, x2)
sage: p2.projective(0)
Point (x1, x0 + x2)
sage: p2.projective(1)
Point (x0, x1 + x2)
sage: p2.projective(2)
Point (x0, x1 + x2)
rational_point()

Return the rational point if this closed point is of degree \(1\).

EXAMPLES:

sage: A.<x,y> = AffineSpace(GF(3^2),2)
sage: C = Curve(y^2 - x^5 - x^4 - 2*x^3 - 2*x-2)
sage: C.closed_points()
[Point (x, y + (z2 + 1)),
 Point (x, y + (-z2 - 1)),
 Point (x + (z2 + 1), y + (z2 - 1)),
 Point (x + (z2 + 1), y + (-z2 + 1)),
 Point (x - 1, y + (z2 + 1)),
 Point (x - 1, y + (-z2 - 1)),
 Point (x + (-z2 - 1), y + z2),
 Point (x + (-z2 - 1), y + (-z2)),
 Point (x + 1, y + 1),
 Point (x + 1, y - 1)]
sage: [p.rational_point() for p in _]
[(0, 2*z2 + 2),
 (0, z2 + 1),
 (2*z2 + 2, 2*z2 + 1),
 (2*z2 + 2, z2 + 2),
 (1, 2*z2 + 2),
 (1, z2 + 1),
 (z2 + 1, 2*z2),
 (z2 + 1, z2),
 (2, 2),
 (2, 1)]
sage: set(_) == set(C.rational_points())
True
class sage.schemes.curves.closed_point.IntegralCurveClosedPoint(curve, prime_ideal, degree)

Bases: sage.schemes.curves.closed_point.CurveClosedPoint

Closed points of integral curves.

INPUT:

  • curve – the curve to which the closed point belongs

  • prime_ideal – a prime ideal

  • degree – degree of the closed point

EXAMPLES:

sage: F.<a> = GF(4)
sage: P.<x,y> = AffineSpace(F, 2);
sage: C = Curve(y^2 + y - x^3)
sage: C.closed_points()
[Point (x, y),
 Point (x, y + 1),
 Point (x + a, y + a),
 Point (x + a, y + (a + 1)),
 Point (x + (a + 1), y + a),
 Point (x + (a + 1), y + (a + 1)),
 Point (x + 1, y + a),
 Point (x + 1, y + (a + 1))]
curve()

Return the curve to which this point belongs.

EXAMPLES:

sage: F.<a> = GF(4)
sage: P.<x,y> = AffineSpace(F, 2);
sage: C = Curve(y^2 + y - x^3)
sage: pts = C.closed_points()
sage: p = pts[0]
sage: p.curve()
Affine Plane Curve over Finite Field in a of size 2^2 defined by x^3 + y^2 + y
degree()

Return the degree of the point.

EXAMPLES:

sage: F.<a> = GF(4)
sage: P.<x,y> = AffineSpace(F, 2);
sage: C = Curve(y^2 + y - x^3)
sage: pts = C.closed_points()
sage: p = pts[0]
sage: p.degree()
1
place()

Return a place on this closed point.

If there are more than one, arbitrary one is chosen.

EXAMPLES:

sage: F.<a> = GF(4)
sage: P.<x,y> = AffineSpace(F, 2);
sage: C = Curve(y^2 + y - x^3)
sage: pts = C.closed_points()
sage: p = pts[0]
sage: p.place()
Place (x, y)
places()

Return all places on this closed point.

EXAMPLES:

sage: F.<a> = GF(4)
sage: P.<x,y> = AffineSpace(F, 2);
sage: C = Curve(y^2 + y - x^3)
sage: pts = C.closed_points()
sage: p = pts[0]
sage: p.places()
[Place (x, y)]
class sage.schemes.curves.closed_point.IntegralProjectiveCurveClosedPoint(curve, prime_ideal, degree)

Bases: sage.schemes.curves.closed_point.IntegralCurveClosedPoint

Closed points of projective plane curves.

affine(i=None)

Return the point in the i-th affine patch of the curve.

INPUT:

  • i – an integer; if not specified, it is chosen automatically.

EXAMPLES:

sage: F.<a> = GF(2)
sage: P.<x,y,z> = ProjectiveSpace(F, 2)
sage: C = Curve(x^3*y + y^3*z + x*z^3)
sage: p1, p2, p3 = C.closed_points()
sage: p1.affine()
Point (x, z)
sage: p2.affine()
Point (x, y)
sage: p3.affine()
Point (y, z)
sage: p3.affine(0)
Point (y, z)
sage: p3.affine(1)
Traceback (most recent call last):
...
ValueError: not in the affine patch
rational_point()

Return the rational point if this closed point is of degree \(1\).

EXAMPLES:

sage: F.<a> = GF(4)
sage: P.<x,y,z> = ProjectiveSpace(F, 2)
sage: C = Curve(x^3*y + y^3*z + x*z^3)
sage: C.closed_points()
[Point (x, z),
 Point (x, y),
 Point (y, z),
 Point (x + a*z, y + (a + 1)*z),
 Point (x + (a + 1)*z, y + a*z)]
sage: [p.rational_point() for p in _]
[(0 : 1 : 0), (0 : 0 : 1), (1 : 0 : 0), (a : a + 1 : 1), (a + 1 : a : 1)]
sage: set(_) == set(C.rational_points())
True