Affine curves

EXAMPLES:

We can construct curves in either an affine plane:

sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = Curve([y - x^2], A); C
Affine Plane Curve over Rational Field defined by -x^2 + y

or in higher dimensional affine space:

sage: A.<x,y,z,w> = AffineSpace(QQ, 4)
sage: C = Curve([y - x^2, z - w^3, w - y^4], A); C
Affine Curve over Rational Field defined by -x^2 + y, -w^3 + z, -y^4 + w

AUTHORS:

  • William Stein (2005-11-13)
  • David Joyner (2005-11-13)
  • David Kohel (2006-01)
  • Grayson Jorgenson (2016-8)
class sage.schemes.curves.affine_curve.AffineCurve(A, X)

Bases: sage.schemes.curves.curve.Curve_generic, sage.schemes.affine.affine_subscheme.AlgebraicScheme_subscheme_affine

Initialization function.

EXAMPLES:

sage: R.<v> = QQ[]
sage: K.<u> = NumberField(v^2 + 3)
sage: A.<x,y,z> = AffineSpace(K, 3)
sage: C = Curve([z - u*x^2, y^2], A); C
Affine Curve over Number Field in u with defining polynomial v^2 + 3
defined by (-u)*x^2 + z, y^2
sage: A.<x,y,z> = AffineSpace(GF(7), 3)
sage: C = Curve([x^2 - z, z - 8*x], A); C
Affine Curve over Finite Field of size 7 defined by x^2 - z, -x + z
blowup(P=None)

Return the blow up of this affine curve at the point P.

The blow up is described by affine charts. This curve must be irreducible.

INPUT:

  • P – (default: None) a point on this curve at which to blow up. If None, then P is taken to be the origin.

OUTPUT:

  • a tuple consisting of three elements. The first is a tuple of curves in affine space of the same dimension as the ambient space of this curve, which define the blow up in each affine chart. The second is a tuple of tuples such that the jth element of the ith tuple is the transition map from the ith affine patch to the jth affine patch. Lastly, the third element is a tuple consisting of the restrictions of the projection map from the blow up back to the original curve, restricted to each affine patch. There the ith element will be the projection from the ith affine patch.

EXAMPLES:

sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = Curve([y^2 - x^3], A)
sage: C.blowup()
((Affine Plane Curve over Rational Field defined by s1^2 - x,
 Affine Plane Curve over Rational Field defined by y*s0^3 - 1),
([Scheme endomorphism of Affine Plane Curve over Rational Field defined by s1^2 - x
    Defn: Defined on coordinates by sending (x, s1) to
          (x, s1), Scheme morphism:
    From: Affine Plane Curve over Rational Field defined by s1^2 - x
    To:   Affine Plane Curve over Rational Field defined by y*s0^3 - 1
    Defn: Defined on coordinates by sending (x, s1) to
          (x*s1, 1/s1)], [Scheme morphism:
    From: Affine Plane Curve over Rational Field defined by y*s0^3 - 1
    To:   Affine Plane Curve over Rational Field defined by s1^2 - x
    Defn: Defined on coordinates by sending (y, s0) to
          (y*s0, 1/s0),
  Scheme endomorphism of Affine Plane Curve over Rational Field defined by y*s0^3 - 1
    Defn: Defined on coordinates by sending (y, s0) to
          (y, s0)]),
(Scheme morphism:
   From: Affine Plane Curve over Rational Field defined by s1^2 - x
   To:   Affine Plane Curve over Rational Field defined by -x^3 + y^2
   Defn: Defined on coordinates by sending (x, s1) to
         (x, x*s1), Scheme morphism:
   From: Affine Plane Curve over Rational Field defined by y*s0^3 - 1
   To:   Affine Plane Curve over Rational Field defined by -x^3 + y^2
   Defn: Defined on coordinates by sending (y, s0) to
         (y*s0, y)))
sage: K.<a> = QuadraticField(2)
sage: A.<x,y,z> = AffineSpace(K, 3)
sage: C = Curve([y^2 - a*x^5, x - z], A)
sage: B = C.blowup()
sage: B[0]
(Affine Curve over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? defined by s2 - 1, 2*x^3 + (-a)*s1^2,
 Affine Curve over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? defined by s0 - s2, 2*y^3*s2^5 + (-a),
 Affine Curve over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? defined by s0 - 1, 2*z^3 + (-a)*s1^2)
sage: B[1][0][2]
Scheme morphism:
  From: Affine Curve over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? defined by s2 - 1, 2*x^3 + (-a)*s1^2
  To:   Affine Curve over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? defined by s0 - 1, 2*z^3 + (-a)*s1^2
  Defn: Defined on coordinates by sending (x, s1, s2) to
        (x*s2, 1/s2, s1/s2)
sage: B[1][2][0]
Scheme morphism:
  From: Affine Curve over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? defined by s0 - 1, 2*z^3 + (-a)*s1^2
  To:   Affine Curve over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? defined by s2 - 1, 2*x^3 + (-a)*s1^2
  Defn: Defined on coordinates by sending (z, s0, s1) to
        (z*s0, s1/s0, 1/s0)
sage: B[2]
(Scheme morphism:
   From: Affine Curve over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? defined by s2 - 1, 2*x^3 + (-a)*s1^2
   To:   Affine Curve over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? defined by (-a)*x^5 + y^2, x - z
   Defn: Defined on coordinates by sending (x, s1, s2) to
         (x, x*s1, x*s2), Scheme morphism:
   From: Affine Curve over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? defined by s0 - s2, 2*y^3*s2^5 + (-a)
   To:   Affine Curve over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? defined by (-a)*x^5 + y^2, x - z
   Defn: Defined on coordinates by sending (y, s0, s2) to
         (y*s0, y, y*s2), Scheme morphism:
   From: Affine Curve over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? defined by s0 - 1, 2*z^3 + (-a)*s1^2
   To:   Affine Curve over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? defined by (-a)*x^5 + y^2, x - z
   Defn: Defined on coordinates by sending (z, s0, s1) to
         (z*s0, z*s1, z))
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = A.curve((y - 3/2)^3 - (x + 2)^5 - (x + 2)^6)
sage: Q = A([-2,3/2])
sage: C.blowup(Q)
((Affine Plane Curve over Rational Field defined by x^3 - s1^3 + 7*x^2 + 16*x + 12,
  Affine Plane Curve over Rational Field defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5 +
  54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8),
 ([Scheme endomorphism of Affine Plane Curve over Rational Field defined by x^3 - s1^3 + 7*x^2 +
 16*x + 12
     Defn: Defined on coordinates by sending (x, s1) to
           (x, s1), Scheme morphism:
     From: Affine Plane Curve over Rational Field defined by x^3 - s1^3 + 7*x^2 + 16*x + 12
     To:   Affine Plane Curve over Rational Field defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5 +
     54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8
     Defn: Defined on coordinates by sending (x, s1) to
           (x*s1 + 2*s1 + 3/2, 1/s1)], [Scheme morphism:
     From: Affine Plane Curve over Rational Field defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5 +
     54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8
     To:   Affine Plane Curve over Rational Field defined by x^3 - s1^3 + 7*x^2 + 16*x + 12
     Defn: Defined on coordinates by sending (y, s0) to
           (y*s0 - 3/2*s0 - 2, 1/s0),
   Scheme endomorphism of Affine Plane Curve over Rational Field defined by 8*y^3*s0^6 - 36*y^2*s0^6 +
   8*y^2*s0^5 + 54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8
     Defn: Defined on coordinates by sending (y, s0) to
           (y, s0)]),
 (Scheme morphism:
    From: Affine Plane Curve over Rational Field defined by x^3 - s1^3 + 7*x^2 + 16*x + 12
    To:   Affine Plane Curve over Rational Field defined by -x^6 - 13*x^5 - 70*x^4 - 200*x^3 + y^3 -
    320*x^2 - 9/2*y^2 - 272*x + 27/4*y - 795/8
    Defn: Defined on coordinates by sending (x, s1) to
          (x, x*s1 + 2*s1 + 3/2), Scheme morphism:
    From: Affine Plane Curve over Rational Field defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5 +
    54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8
    To:   Affine Plane Curve over Rational Field defined by -x^6 - 13*x^5 - 70*x^4 - 200*x^3 + y^3 -
    320*x^2 - 9/2*y^2 - 272*x + 27/4*y - 795/8
    Defn: Defined on coordinates by sending (y, s0) to
          (y*s0 - 3/2*s0 - 2, y)))
sage: A.<x,y,z,w> = AffineSpace(QQ, 4)
sage: C = A.curve([((x + 1)^2 + y^2)^3 - 4*(x + 1)^2*y^2, y - z, w - 4])
sage: Q = C([-1,0,0,4])
sage: B = C.blowup(Q)
sage: B[0]
(Affine Curve over Rational Field defined by s3, s1 - s2, x^2*s2^6 +
2*x*s2^6 + 3*x^2*s2^4 + s2^6 + 6*x*s2^4 + 3*x^2*s2^2 + 3*s2^4 + 6*x*s2^2
+ x^2 - s2^2 + 2*x + 1,
 Affine Curve over Rational Field defined by s3, s2 - 1, y^2*s0^6 +
3*y^2*s0^4 + 3*y^2*s0^2 + y^2 - 4*s0^2,
 Affine Curve over Rational Field defined by s3, s1 - 1, z^2*s0^6 +
3*z^2*s0^4 + 3*z^2*s0^2 + z^2 - 4*s0^2,
 Closed subscheme of Affine Space of dimension 4 over Rational Field
defined by:
   1)
sage: Q = A([6,2,3,1])
sage: B = C.blowup(Q)
Traceback (most recent call last):
...
TypeError: (=(6, 2, 3, 1)) must be a point on this curve
sage: A.<x,y> = AffineSpace(QuadraticField(-1), 2)
sage: C = A.curve([y^2 + x^2])
sage: C.blowup()
Traceback (most recent call last):
...
TypeError: this curve must be irreducible
plane_projection(AP=None)

Return a projection of this curve into an affine plane so that the image of the projection is a plane curve.

INPUT:

  • AP – (default: None) the affine plane to project this curve into. This space must be defined over the same base field as this curve, and must have dimension two. This space will be constructed if not specified.

OUTPUT:

  • a tuple consisting of two elements: a scheme morphism from this curve into an affine plane, and the plane curve that defines the image of that morphism.

EXAMPLES:

sage: A.<x,y,z,w> = AffineSpace(QQ, 4)
sage: C = Curve([x^2 - y*z*w, z^3 - w, w + x*y - 3*z^3], A)
sage: C.plane_projection()
(Scheme morphism:
  From: Affine Curve over Rational Field defined by -y*z*w + x^2, z^3 -
w, -3*z^3 + x*y + w
  To:   Affine Space of dimension 2 over Rational Field
  Defn: Defined on coordinates by sending (x, y, z, w) to
        (x, y), Affine Plane Curve over Rational Field defined by
x0^2*x1^7 - 16*x0^4)
sage: R.<a> = QQ[]
sage: K.<b> = NumberField(a^2 + 2)
sage: A.<x,y,z> = AffineSpace(K, 3)
sage: C = A.curve([x - b, y - 2])
sage: B.<a,b> = AffineSpace(K, 2)
sage: proj1 = C.plane_projection(AP=B)
sage: proj1
(Scheme morphism:
   From: Affine Curve over Number Field in b with defining polynomial
a^2 + 2 defined by x + (-b), y - 2
   To:   Affine Space of dimension 2 over Number Field in b with
defining polynomial a^2 + 2
   Defn: Defined on coordinates by sending (x, y, z) to
         (x, z),
 Affine Plane Curve over Number Field in b with defining polynomial a^2
+ 2 defined by a + (-b))
sage: proj1[1].ambient_space() is B
True
sage: proj2 = C.plane_projection()
sage: proj2[1].ambient_space() is B
False
projection(indices, AS=None)

Return the projection of this curve onto the coordinates specified by indices.

INPUT:

  • indices – a list or tuple of distinct integers specifying the indices of the coordinates to use in the projection. Can also be a list or tuple consisting of variables of the coordinate ring of the ambient space of this curve. If integers are used to specify the coordinates, 0 denotes the first coordinate. The length of indices must be between two and one less than the dimension of the ambient space of this curve, inclusive.
  • AS – (default: None) the affine space the projected curve will be defined in. This space must be defined over the same base field as this curve, and must have dimension equal to the length of indices. This space is constructed if not specified.

OUTPUT:

  • a tuple consisting of two elements: a scheme morphism from this curve to affine space of dimension equal to the number of coordinates specified in indices, and the affine subscheme that is the image of that morphism. If the image is a curve, the second element of the tuple will be a curve.

EXAMPLES:

sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: C = Curve([y^7 - x^2 + x^3 - 2*z, z^2 - x^7 - y^2], A)
sage: C.projection([0,1])
(Scheme morphism:
   From: Affine Curve over Rational Field defined by y^7 + x^3 - x^2 -
2*z, -x^7 - y^2 + z^2
   To:   Affine Space of dimension 2 over Rational Field
   Defn: Defined on coordinates by sending (x, y, z) to
         (x, y),
 Affine Plane Curve over Rational Field defined by x1^14 + 2*x0^3*x1^7 -
2*x0^2*x1^7 - 4*x0^7 + x0^6 - 2*x0^5 + x0^4 - 4*x1^2)
sage: C.projection([0,1,3,4])
Traceback (most recent call last):
...
ValueError: (=[0, 1, 3, 4]) must be a list or tuple of length between 2
and (=2), inclusive
sage: A.<x,y,z,w> = AffineSpace(QQ, 4)
sage: C = Curve([x - 2, y - 3, z - 1], A)
sage: B.<a,b,c> = AffineSpace(QQ, 3)
sage: C.projection([0,1,2], AS=B)
(Scheme morphism:
   From: Affine Curve over Rational Field defined by x - 2, y - 3, z - 1
   To:   Affine Space of dimension 3 over Rational Field
   Defn: Defined on coordinates by sending (x, y, z, w) to
         (x, y, z),
 Closed subscheme of Affine Space of dimension 3 over Rational Field
defined by:
   c - 1,
   b - 3,
   a - 2)
sage: A.<x,y,z,w,u> = AffineSpace(GF(11), 5)
sage: C = Curve([x^3 - 5*y*z + u^2, x - y^2 + 3*z^2, w^2 + 2*u^3*y, y - u^2 + z*x], A)
sage: B.<a,b,c> = AffineSpace(GF(11), 3)
sage: proj1 = C.projection([1,2,4], AS=B)
sage: proj1
(Scheme morphism:
   From: Affine Curve over Finite Field of size 11 defined by x^3 -
5*y*z + u^2, -y^2 + 3*z^2 + x, 2*y*u^3 + w^2, x*z - u^2 + y
   To:   Affine Space of dimension 3 over Finite Field of size 11
   Defn: Defined on coordinates by sending (x, y, z, w, u) to
         (y, z, u),
 Affine Curve over Finite Field of size 11 defined by a^2*b - 3*b^3 -
c^2 + a, c^6 - 5*a*b^4 + b^3*c^2 - 3*a*c^4 + 3*a^2*c^2 - a^3, a^2*c^4 -
3*b^2*c^4 - 2*a^3*c^2 - 5*a*b^2*c^2 + a^4 - 5*a*b^3 + 2*b^4 + b^2*c^2 -
3*b*c^2 + 3*a*b, a^4*c^2 + 2*b^4*c^2 - a^5 - 2*a*b^4 + 5*b*c^4 + a*b*c^2
- 5*a*b^2 + 4*b^3 + b*c^2 + 5*c^2 - 5*a, a^6 - 5*b^6 - 5*b^3*c^2 +
5*a*b^3 + 2*c^4 - 4*a*c^2 + 2*a^2 - 5*a*b + c^2)
sage: proj1[1].ambient_space() is B
True
sage: proj2 = C.projection([1,2,4])
sage: proj2[1].ambient_space() is B
False
sage: C.projection([1,2,3,5], AS=B)
Traceback (most recent call last):
...
TypeError: (=Affine Space of dimension 3 over Finite Field of size 11)
must have dimension (=4)
sage: A.<x,y,z,w> = AffineSpace(QQ, 4)
sage: C = A.curve([x*y - z^3, x*z - w^3, w^2 - x^3])
sage: C.projection([y,z])
(Scheme morphism:
   From: Affine Curve over Rational Field defined by -z^3 + x*y, -w^3 +
x*z, -x^3 + w^2
   To:   Affine Space of dimension 2 over Rational Field
   Defn: Defined on coordinates by sending (x, y, z, w) to
         (y, z),
 Affine Plane Curve over Rational Field defined by x1^23 - x0^7*x1^4)
sage: B.<x,y,z> = AffineSpace(QQ, 3)
sage: C.projection([x,y,z], AS=B)
(Scheme morphism:
   From: Affine Curve over Rational Field defined by -z^3 + x*y, -w^3 +
x*z, -x^3 + w^2
   To:   Affine Space of dimension 3 over Rational Field
   Defn: Defined on coordinates by sending (x, y, z, w) to
         (x, y, z),
 Affine Curve over Rational Field defined by z^3 - x*y, x^8 - x*z^2,
x^7*z^2 - x*y*z)
sage: C.projection([y,z,z])
Traceback (most recent call last):
...
ValueError: (=[y, z, z]) must be a list or tuple of distinct indices or
variables
projective_closure(i=0, PP=None)

Return the projective closure of this affine curve.

INPUT:

  • i – (default: 0) the index of the affine coordinate chart of the projective space that the affine ambient space of this curve embeds into.
  • PP – (default: None) ambient projective space to compute the projective closure in. This is constructed if it is not given.

OUTPUT:

  • a curve in projective space.

EXAMPLES:

sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: C = Curve([y-x^2,z-x^3], A)
sage: C.projective_closure()
Projective Curve over Rational Field defined by x1^2 - x0*x2,
x1*x2 - x0*x3, x2^2 - x1*x3
sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: C = Curve([y - x^2, z - x^3], A)
sage: C.projective_closure()
Projective Curve over Rational Field defined by
x1^2 - x0*x2, x1*x2 - x0*x3, x2^2 - x1*x3
sage: A.<x,y> = AffineSpace(CC, 2)
sage: C = Curve(y - x^3 + x - 1, A)
sage: C.projective_closure(1)
Projective Plane Curve over Complex Field with 53 bits of precision defined by
x0^3 - x0*x1^2 + x1^3 - x1^2*x2
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: P.<u,v,w> = ProjectiveSpace(QQ, 2)
sage: C = Curve([y - x^2], A)
sage: C.projective_closure(1, P).ambient_space() == P
True
resolution_of_singularities(extend=False)

Return a nonsingular model for this affine curve created by blowing up its singular points.

The nonsingular model is given as a collection of affine patches that cover it. If extend is False and if the base field is a number field, or if the base field is a finite field, the model returned may have singularities with coordinates not contained in the base field. An error is returned if this curve is already nonsingular, or if it has no singular points over its base field. This curve must be irreducible, and must be defined over a number field or finite field.

INPUT:

  • extend – (default: False) specifies whether to extend the base field when necessary to find all singular points when this curve is defined over a number field. If extend is False, then only singularities with coordinates in the base field of this curve will be resolved. However, setting extend to True will slow down computations.

OUTPUT:

  • a tuple consisting of three elements. The first is a tuple of curves in affine space of the same dimension as the ambient space of this curve, which represent affine patches of the resolution of singularities. The second is a tuple of tuples such that the jth element of the ith tuple is the transition map from the ith patch to the jth patch. Lastly, the third element is a tuple consisting of birational maps from the patches back to the original curve that were created by composing the projection maps generated from the blow up computations. There the ith element will be a map from the ith patch.

EXAMPLES:

sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = Curve([y^2 - x^3], A)
sage: C.resolution_of_singularities()
((Affine Plane Curve over Rational Field defined by s1^2 - x,
  Affine Plane Curve over Rational Field defined by y*s0^3 - 1),
 ((Scheme endomorphism of Affine Plane Curve over Rational Field defined by s1^2 - x
     Defn: Defined on coordinates by sending (x, s1) to
           (x, s1), Scheme morphism:
     From: Affine Plane Curve over Rational Field defined by s1^2 - x
     To:   Affine Plane Curve over Rational Field defined by y*s0^3 - 1
     Defn: Defined on coordinates by sending (x, s1) to
           (x*s1, 1/s1)), (Scheme morphism:
     From: Affine Plane Curve over Rational Field defined by y*s0^3 - 1
     To:   Affine Plane Curve over Rational Field defined by s1^2 - x
     Defn: Defined on coordinates by sending (y, s0) to
           (y*s0, 1/s0),
   Scheme endomorphism of Affine Plane Curve over Rational Field defined by y*s0^3 - 1
     Defn: Defined on coordinates by sending (y, s0) to
           (y, s0))),
 (Scheme morphism:
    From: Affine Plane Curve over Rational Field defined by s1^2 - x
    To:   Affine Plane Curve over Rational Field defined by -x^3 + y^2
    Defn: Defined on coordinates by sending (x, s1) to
          (x, x*s1), Scheme morphism:
    From: Affine Plane Curve over Rational Field defined by y*s0^3 - 1
    To:   Affine Plane Curve over Rational Field defined by -x^3 + y^2
    Defn: Defined on coordinates by sending (y, s0) to
          (y*s0, y)))
sage: set_verbose(-1)
sage: K.<a> = QuadraticField(3)
sage: A.<x,y> = AffineSpace(K, 2)
sage: C = A.curve(x^4 + 2*x^2 + a*y^3 + 1)
sage: C.resolution_of_singularities(extend=True)[0] # long time (2 seconds)
(Affine Plane Curve over Number Field in a0 with defining polynomial y^4 - 4*y^2 + 16 defined by
24*x^2*ss1^3 + 24*ss1^3 + (a0^3 - 8*a0),
 Affine Plane Curve over Number Field in a0 with defining polynomial y^4 - 4*y^2 + 16 defined by
 24*s1^2*ss0 + (a0^3 - 8*a0)*ss0^2 + (-6*a0^3)*s1,
 Affine Plane Curve over Number Field in a0 with defining polynomial y^4 - 4*y^2 + 16 defined by
 8*y^2*s0^4 + (4*a0^3)*y*s0^3 - 32*s0^2 + (a0^3 - 8*a0)*y)
sage: A.<x,y,z> = AffineSpace(GF(5), 3)
sage: C = Curve([y - x^3, (z - 2)^2 - y^3 - x^3], A)
sage: R = C.resolution_of_singularities()
sage: R[0]
(Affine Curve over Finite Field of size 5 defined by x^2 - s1, s1^4 - x*s2^2 + s1, x*s1^3 - s2^2 + x,
 Affine Curve over Finite Field of size 5 defined by y*s2^2 - y^2 - 1, s2^4 - s0^3 - y^2 - 2, y*s0^3
 - s2^2 + y, Affine Curve over Finite Field of size 5 defined by s0^3*s1 + z*s1^3 + s1^4 - 2*s1^3 - 1,
 z*s0^3 + z*s1^3 - 2*s0^3 - 2*s1^3 - 1, z^2*s1^3 + z*s1^3 - s1^3 - z + s1 + 2)
sage: A.<x,y,z,w> = AffineSpace(QQ, 4)
sage: C = A.curve([((x - 2)^2 + y^2)^2 - (x - 2)^2 - y^2 + (x - 2)^3, z - y - 7, w - 4])
sage: B = C.resolution_of_singularities()
sage: B[0]
(Affine Curve over Rational Field defined by s3, s1 - s2, x^2*s2^4 -
4*x*s2^4 + 2*x^2*s2^2 + 4*s2^4 - 8*x*s2^2 + x^2 + 7*s2^2 - 3*x + 1,
 Affine Curve over Rational Field defined by s3, s2 - 1, y^2*s0^4 +
2*y^2*s0^2 + y*s0^3 + y^2 - s0^2 - 1,
 Affine Curve over Rational Field defined by s3, s1 - 1, z^2*s0^4 -
14*z*s0^4 + 2*z^2*s0^2 + z*s0^3 + 49*s0^4 - 28*z*s0^2 - 7*s0^3 + z^2 +
97*s0^2 - 14*z + 48,
 Closed subscheme of Affine Space of dimension 4 over Rational Field
defined by:
   1)
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = Curve([y - x^2 + 1], A)
sage: C.resolution_of_singularities()
Traceback (most recent call last):
...
TypeError: this curve is already nonsingular
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = A.curve([(x^2 + y^2 - y - 2)*(y - x^2 + 2) + y^3])
sage: C.resolution_of_singularities()
Traceback (most recent call last):
...
TypeError: this curve has no singular points over its base field. If
working over a number field use extend=True
class sage.schemes.curves.affine_curve.AffinePlaneCurve(A, f)

Bases: sage.schemes.curves.affine_curve.AffineCurve

Initialization function.

EXAMPLES:

sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = Curve([x^3 - y^2], A); C
Affine Plane Curve over Rational Field defined by x^3 - y^2
sage: A.<x,y> = AffineSpace(CC, 2)
sage: C = Curve([y^2 + x^2], A); C
Affine Plane Curve over Complex Field with 53 bits of precision defined
by x^2 + y^2
divisor_of_function(r)

Return the divisor of a function on a curve.

INPUT: r is a rational function on X

OUTPUT:

  • list - The divisor of r represented as a list of coefficients and points. (TODO: This will change to a more structural output in the future.)

EXAMPLES:

sage: F = GF(5)
sage: P2 = AffineSpace(2, F, names = 'xy')
sage: R = P2.coordinate_ring()
sage: x, y = R.gens()
sage: f = y^2 - x^9 - x
sage: C = Curve(f)
sage: K = FractionField(R)
sage: r = 1/x
sage: C.divisor_of_function(r)     # todo: not implemented (broken)
      [[-1, (0, 0, 1)]]
sage: r = 1/x^3
sage: C.divisor_of_function(r)     # todo: not implemented (broken)
      [[-3, (0, 0, 1)]]
fundamental_group()

Return a presentation of the fundamental group of the complement of self.

Note

The curve must be defined over the rationals or a number field with an embedding over \(\QQbar\).

EXAMPLES:

sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = A.curve(y^2 - x^3 - x^2)
sage: C.fundamental_group() # optional - sirocco
Finitely presented group < x0 |  >

In the case of number fields, they need to have an embedding to the algebraic field:

sage: a = QQ[x](x^2+5).roots(QQbar)[0][0]
sage: F = NumberField(a.minpoly(), 'a', embedding=a)
sage: F.inject_variables()
Defining a
sage: A.<x,y> = AffineSpace(F, 2)
sage: C = A.curve(y^2 - a*x^3 - x^2)
sage: C.fundamental_group() # optional - sirocco
Finitely presented group < x0 |  >

Warning

This functionality requires the sirocco package to be installed.

is_ordinary_singularity(P)

Return whether the singular point P of this affine plane curve is an ordinary singularity.

The point P is an ordinary singularity of this curve if it is a singular point, and if the tangents of this curve at P are distinct.

INPUT:

  • P – a point on this curve.

OUTPUT:

  • Boolean. True or False depending on whether P is or is not an ordinary singularity of this curve, respectively. An error is raised if P is not a singular point of this curve.

EXAMPLES:

sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = Curve([y^2 - x^3], A)
sage: Q = A([0,0])
sage: C.is_ordinary_singularity(Q)
False
sage: R.<a> = QQ[]
sage: K.<b> = NumberField(a^2 - 3)
sage: A.<x,y> = AffineSpace(K, 2)
sage: C = Curve([(x^2 + y^2 - 2*x)^2 - x^2 - y^2], A)
sage: Q = A([0,0])
sage: C.is_ordinary_singularity(Q)
True
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = A.curve([x^2*y - y^2*x + y^2 + x^3])
sage: Q = A([-1,-1])
sage: C.is_ordinary_singularity(Q)
Traceback (most recent call last):
...
TypeError: (=(-1, -1)) is not a singular point of (=Affine Plane Curve
over Rational Field defined by x^3 + x^2*y - x*y^2 + y^2)
is_transverse(C, P)

Return whether the intersection of this curve with the curve C at the point P is transverse.

The intersection at P is transverse if P is a nonsingular point of both curves, and if the tangents of the curves at P are distinct.

INPUT:

  • C – a curve in the ambient space of this curve.
  • P – a point in the intersection of both curves.

OUTPUT: Boolean.

EXAMPLES:

sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = Curve([x^2 + y^2 - 1], A)
sage: D = Curve([x - 1], A)
sage: Q = A([1,0])
sage: C.is_transverse(D, Q)
False
sage: R.<a> = QQ[]
sage: K.<b> = NumberField(a^3 + 2)
sage: A.<x,y> = AffineSpace(K, 2)
sage: C = A.curve([x*y])
sage: D = A.curve([y - b*x])
sage: Q = A([0,0])
sage: C.is_transverse(D, Q)
False
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = Curve([y - x^3], A)
sage: D = Curve([y + x], A)
sage: Q = A([0,0])
sage: C.is_transverse(D, Q)
True
local_coordinates(pt, n)

Return local coordinates to precision n at the given point.

Behaviour is flaky - some choices of \(n\) are worst that others.

INPUT:

  • pt - an F-rational point on X which is not a point of ramification for the projection (x,y) - x.
  • n - the number of terms desired

OUTPUT: x = x0 + t y = y0 + power series in t

EXAMPLES:

sage: F = GF(5)
sage: pt = (2,3)
sage: R = PolynomialRing(F,2, names = ['x','y'])
sage: x,y = R.gens()
sage: f = y^2-x^9-x
sage: C = Curve(f)
sage: C.local_coordinates(pt, 9)
[t + 2, -2*t^12 - 2*t^11 + 2*t^9 + t^8 - 2*t^7 - 2*t^6 - 2*t^4 + t^3 - 2*t^2 - 2]
multiplicity(P)

Return the multiplicity of this affine plane curve at the point P.

In the special case of affine plane curves, the multiplicity of an affine plane curve at the point (0,0) can be computed as the minimum of the degrees of the homogeneous components of its defining polynomial. To compute the multiplicity of a different point, a linear change of coordinates is used.

This curve must be defined over a field. An error if raised if P is not a point on this curve.

INPUT:

  • P – a point in the ambient space of this curve.

OUTPUT:

An integer.

EXAMPLES:

sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = Curve([y^2 - x^3], A)
sage: Q1 = A([1,1])
sage: C.multiplicity(Q1)
1
sage: Q2 = A([0,0])
sage: C.multiplicity(Q2)
2
sage: A.<x,y> = AffineSpace(QQbar,2)
sage: C = Curve([-x^7 + (-7)*x^6 + y^6 + (-21)*x^5 + 12*y^5 + (-35)*x^4 + 60*y^4 +\
(-35)*x^3 + 160*y^3 + (-21)*x^2 + 240*y^2 + (-7)*x + 192*y + 63], A)
sage: Q = A([-1,-2])
sage: C.multiplicity(Q)
6
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = A.curve([y^3 - x^3 + x^6])
sage: Q = A([1,1])
sage: C.multiplicity(Q)
Traceback (most recent call last):
...
TypeError: (=(1, 1)) is not a point on (=Affine Plane Curve over
Rational Field defined by x^6 - x^3 + y^3)
plot(*args, **kwds)

Plot the real points on this affine plane curve.

INPUT:

  • self - an affine plane curve
  • *args - optional tuples (variable, minimum, maximum) for plotting dimensions
  • **kwds - optional keyword arguments passed on to implicit_plot

EXAMPLES:

A cuspidal curve:

sage: R.<x, y> = QQ[]
sage: C = Curve(x^3 - y^2)
sage: C.plot()
Graphics object consisting of 1 graphics primitive

A 5-nodal curve of degree 11. This example also illustrates some of the optional arguments:

sage: R.<x, y> = ZZ[]
sage: C = Curve(32*x^2 - 2097152*y^11 + 1441792*y^9 - 360448*y^7 + 39424*y^5 - 1760*y^3 + 22*y - 1)
sage: C.plot((x, -1, 1), (y, -1, 1), plot_points=400)
Graphics object consisting of 1 graphics primitive

A line over \(\mathbf{RR}\):

sage: R.<x, y> = RR[]
sage: C = Curve(R(y - sqrt(2)*x))
sage: C.plot()
Graphics object consisting of 1 graphics primitive
rational_parameterization()

Return a rational parameterization of this curve.

This curve must have rational coefficients and be absolutely irreducible (i.e. irreducible over the algebraic closure of the rational field). The curve must also be rational (have geometric genus zero).

The rational parameterization may have coefficients in a quadratic extension of the rational field.

OUTPUT:

  • a birational map between \(\mathbb{A}^{1}\) and this curve, given as a scheme morphism.

EXAMPLES:

sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = Curve([y^2 - x], A)
sage: C.rational_parameterization()
Scheme morphism:
  From: Affine Space of dimension 1 over Rational Field
  To:   Affine Plane Curve over Rational Field defined by y^2 - x
  Defn: Defined on coordinates by sending (t) to
        (t^2, t)
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = Curve([(x^2 + y^2 - 2*x)^2 - x^2 - y^2], A)
sage: C.rational_parameterization()
Scheme morphism:
  From: Affine Space of dimension 1 over Rational Field
  To:   Affine Plane Curve over Rational Field defined by x^4 +
2*x^2*y^2 + y^4 - 4*x^3 - 4*x*y^2 + 3*x^2 - y^2
  Defn: Defined on coordinates by sending (t) to
        ((-12*t^4 + 6*t^3 + 4*t^2 - 2*t)/(-25*t^4 + 40*t^3 - 26*t^2 +
8*t - 1), (-9*t^4 + 12*t^3 - 4*t + 1)/(-25*t^4 + 40*t^3 - 26*t^2 + 8*t - 1))
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = Curve([x^2 + y^2 + 7], A)
sage: C.rational_parameterization()
Scheme morphism:
  From: Affine Space of dimension 1 over Number Field in a with defining polynomial a^2 + 7
  To:   Affine Plane Curve over Number Field in a with defining
polynomial a^2 + 7 defined by x^2 + y^2 + 7
  Defn: Defined on coordinates by sending (t) to
        ((-7*t^2 + 7)/((-a)*t^2 + (-a)), 14*t/((-a)*t^2 + (-a)))
riemann_surface(**kwargs)

Return the complex riemann surface determined by this curve

OUTPUT:

  • RiemannSurface object

EXAMPLES:

sage: R.<x,y>=QQ[]
sage: C=Curve(x^3+3*y^3+5)
sage: C.riemann_surface()
Riemann surface defined by polynomial f = x^3 + 3*y^3 + 5 = 0, with 53 bits of precision
tangents(P, factor=True)

Return the tangents of this affine plane curve at the point P.

The point P must be a point on this curve.

INPUT:

  • P – a point on this curve.
  • factor – (default: True) whether to attempt computing the polynomials of the individual tangent lines over the base field of this curve, or to just return the polynomial corresponding to the union of the tangent lines (which requires fewer computations).

OUTPUT:

  • a list of polynomials in the coordinate ring of the ambient space of this curve.

EXAMPLES:

sage: set_verbose(-1)
sage: A.<x,y> = AffineSpace(QQbar, 2)
sage: C = Curve([x^5*y^3 + 2*x^4*y^4 + x^3*y^5 + 3*x^4*y^3 + 6*x^3*y^4 + 3*x^2*y^5\
+ 3*x^3*y^3 + 6*x^2*y^4 + 3*x*y^5 + x^5 + 10*x^4*y + 40*x^3*y^2 + 81*x^2*y^3 + 82*x*y^4\
+ 33*y^5], A)
sage: Q = A([0,0])
sage: C.tangents(Q)
[x + 3.425299577684700?*y, x + (1.949159013086856? + 1.179307909383728?*I)*y,
x + (1.949159013086856? - 1.179307909383728?*I)*y, x + (1.338191198070795? + 0.2560234251008043?*I)*y,
x + (1.338191198070795? - 0.2560234251008043?*I)*y]
sage: C.tangents(Q, factor=False)
[120*x^5 + 1200*x^4*y + 4800*x^3*y^2 + 9720*x^2*y^3 + 9840*x*y^4 + 3960*y^5]
sage: R.<a> = QQ[]
sage: K.<b> = NumberField(a^2 - 3)
sage: A.<x,y> = AffineSpace(K, 2)
sage: C = Curve([(x^2 + y^2 - 2*x)^2 - x^2 - y^2], A)
sage: Q = A([0,0])
sage: C.tangents(Q)
[x + (-1/3*b)*y, x + (1/3*b)*y]
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = A.curve([y^2 - x^3 - x^2])
sage: Q = A([0,0])
sage: C.tangents(Q)
[x - y, x + y]
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = A.curve([y*x - x^4 + 2*x^2])
sage: Q = A([1,1])
sage: C.tangents(Q)
Traceback (most recent call last):
...
TypeError: (=(1, 1)) is not a point on (=Affine Plane Curve over
Rational Field defined by -x^4 + 2*x^2 + x*y)
class sage.schemes.curves.affine_curve.AffinePlaneCurve_finite_field(A, f)

Bases: sage.schemes.curves.affine_curve.AffinePlaneCurve

rational_points(algorithm='enum')

Return sorted list of all rational points on this curve.

Use very naive point enumeration to find all rational points on this curve over a finite field.

EXAMPLES:

sage: A.<x,y> = AffineSpace(2,GF(9,'a'))
sage: C = Curve(x^2 + y^2 - 1)
sage: C
Affine Plane Curve over Finite Field in a of size 3^2 defined by x^2 + y^2 - 1
sage: C.rational_points()
[(0, 1), (0, 2), (1, 0), (2, 0), (a + 1, a + 1), (a + 1, 2*a + 2), (2*a + 2, a + 1), (2*a + 2, 2*a + 2)]
class sage.schemes.curves.affine_curve.AffinePlaneCurve_prime_finite_field(A, f)

Bases: sage.schemes.curves.affine_curve.AffinePlaneCurve_finite_field

rational_points(algorithm='enum')

Return sorted list of all rational points on this curve.

INPUT:

  • algorithm - string:
    • 'enum' - straightforward enumeration
    • 'bn' - via Singular’s Brill-Noether package.
    • 'all' - use all implemented algorithms and verify that they give the same answer, then return it

Note

The Brill-Noether package does not always work. When it fails a RuntimeError exception is raised.

EXAMPLES:

sage: x, y = (GF(5)['x,y']).gens()
sage: f = y^2 - x^9 - x
sage: C = Curve(f); C
Affine Plane Curve over Finite Field of size 5 defined by -x^9 + y^2 - x
sage: C.rational_points(algorithm='bn')
[(0, 0), (2, 2), (2, 3), (3, 1), (3, 4)]
sage: C = Curve(x - y + 1)
sage: C.rational_points()
[(0, 1), (1, 2), (2, 3), (3, 4), (4, 0)]

We compare Brill-Noether and enumeration:

sage: x, y = (GF(17)['x,y']).gens()
sage: C = Curve(x^2 + y^5 + x*y - 19)
sage: v = C.rational_points(algorithm='bn')
sage: w = C.rational_points(algorithm='enum')
sage: len(v)
20
sage: v == w
True
riemann_roch_basis(D)

Interfaces with Singular’s BrillNoether command.

INPUT:

  • self - a plane curve defined by a polynomial eqn f(x,y) = 0 over a prime finite field F = GF(p) in 2 variables x,y representing a curve X: f(x,y) = 0 having n F-rational points (see the Sage function places_on_curve)
  • D - an n-tuple of integers \((d1, ..., dn)\) representing the divisor \(Div = d1*P1+...+dn*Pn\), where \(X(F) = \{P1,...,Pn\}\). The ordering is that dictated by places_on_curve.

OUTPUT: basis of L(Div)

EXAMPLES:

sage: R = PolynomialRing(GF(5),2,names = ["x","y"])
sage: x, y = R.gens()
sage: f = y^2 - x^9 - x
sage: C = Curve(f)
sage: D = [6,0,0,0,0,0]
sage: C.riemann_roch_basis(D)
[1, (-x*z^5 + y^2*z^4)/x^6, (-x*z^6 + y^2*z^5)/x^7, (-x*z^7 + y^2*z^6)/x^8]