Projective curves#
Projective curves in Sage are curves in a projective space or a projective plane.
EXAMPLES:
We can construct curves in either a projective plane:
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: C = Curve([y*z^2 - x^3], P); C
Projective Plane Curve over Rational Field defined by -x^3 + y*z^2
or in higher dimensional projective spaces:
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
sage: C = Curve([y*w^3 - x^4, z*w^3 - x^4], P); C
Projective Curve over Rational Field defined by -x^4 + y*w^3, -x^4 + z*w^3
Integral projective curves over finite fields#
If the curve is defined over a finite field and integral, that is reduced and irreducible, its function field is tightly coupled with the curve so that advanced computations based on Sage’s global function field machinery are available.
EXAMPLES:
sage: k = GF(2) # optional - sage.rings.finite_rings
sage: P.<x,y,z> = ProjectiveSpace(k, 2) # optional - sage.rings.finite_rings
sage: C = Curve(x^2*z - y^3, P) # optional - sage.rings.finite_rings
sage: C.genus() # optional - sage.rings.finite_rings
0
sage: C.function_field() # optional - sage.rings.finite_rings
Function field in z defined by z + y^3
Closed points of arbitrary degree can be computed:
sage: C.closed_points() # optional - sage.rings.finite_rings
[Point (x, y), Point (y, z), Point (x + z, y + z)]
sage: C.closed_points(2) # optional - sage.rings.finite_rings
[Point (y^2 + y*z + z^2, x + z)]
sage: C.closed_points(3) # optional - sage.rings.finite_rings
[Point (y^3 + y^2*z + z^3, x + y + z),
Point (x^2 + y*z + z^2, x*y + x*z + y*z, y^2 + x*z + y*z + z^2)]
All singular closed points can be found:
sage: C.singular_closed_points() # optional - sage.rings.finite_rings
[Point (x, y)]
sage: p = _[0] # optional - sage.rings.finite_rings
sage: p.places() # a unibranch singularity, that is, a cusp # optional - sage.rings.finite_rings
[Place (1/y)]
sage: pls = _[0] # optional - sage.rings.finite_rings
sage: C.place_to_closed_point(pls) # optional - sage.rings.finite_rings
Point (x, y)
It is easy to transit to and from the function field of the curve:
sage: fx = C(x/z) # optional - sage.rings.finite_rings
sage: fy = C(y/z) # optional - sage.rings.finite_rings
sage: fx^2 - fy^3 # optional - sage.rings.finite_rings
0
sage: fx.divisor() # optional - sage.rings.finite_rings
3*Place (1/y)
- 3*Place (y)
sage: p, = fx.poles() # optional - sage.rings.finite_rings
sage: p # optional - sage.rings.finite_rings
Place (y)
sage: C.place_to_closed_point(p) # optional - sage.rings.finite_rings
Point (y, z)
sage: _.rational_point() # optional - sage.rings.finite_rings
(1 : 0 : 0)
sage: _.closed_point() # optional - sage.rings.finite_rings
Point (y, z)
sage: _.place() # optional - sage.rings.finite_rings
Place (y)
Integral projective curves over \(\QQ\)#
An integral curve over \(\QQ\) is also equipped with the function field. Unlike over finite fields, it is not possible to enumerate closed points.
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: C = Curve(x^2*z^2 - x^4 - y^4, P)
sage: C.singular_closed_points()
[Point (x, y)]
sage: p, = _
sage: p.places()
[Place (1/y, 1/y^2*z - 1), Place (1/y, 1/y^2*z + 1)]
sage: fy = C.function(y/z)
sage: fy.divisor()
Place (1/y, 1/y^2*z - 1)
+ Place (1/y, 1/y^2*z + 1)
+ Place (y, z - 1)
+ Place (y, z + 1)
- Place (y^4 + 1, z)
sage: supp = _.support()
sage: pl = supp[0]
sage: C.place_to_closed_point(pl)
Point (x, y)
sage: pl = supp[1]
sage: C.place_to_closed_point(pl)
Point (x, y)
sage: _.rational_point()
(0 : 0 : 1)
sage: _ in C
True
AUTHORS:
William Stein (2005-11-13)
David Joyner (2005-11-13)
David Kohel (2006-01)
Moritz Minzlaff (2010-11)
Grayson Jorgenson (2016-08)
Kwankyu Lee (2019-05): added integral projective curves
- sage.schemes.curves.projective_curve.Hasse_bounds(q, genus=1)#
Return the Hasse-Weil bounds for the cardinality of a nonsingular curve defined over \(\GF{q}\) of given
genus.INPUT:
q(int) – a prime powergenus(int, default 1) – a non-negative integer,
OUTPUT: A tuple. The Hasse bounds (lb,ub) for the cardinality of a curve of genus
genusdefined over \(\GF{q}\).EXAMPLES:
sage: Hasse_bounds(2) (1, 5) sage: Hasse_bounds(next_prime(10^30)) (999999999999998000000000000058, 1000000000000002000000000000058)
- class sage.schemes.curves.projective_curve.IntegralProjectiveCurve(A, f)#
Bases:
ProjectiveCurve_fieldIntegral projective curve.
- coordinate_functions(i=None)#
Return the coordinate functions for the
i-th affine patch.If
iisNone, return the homogeneous coordinate functions.EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(GF(4), 2) # optional - sage.rings.finite_rings sage: C = Curve(x^5 + y^5 + x*y*z^3 + z^5) # optional - sage.rings.finite_rings sage: C.coordinate_functions(0) # optional - sage.rings.finite_rings (y, z) sage: C.coordinate_functions(1) # optional - sage.rings.finite_rings (1/y, 1/y*z)
- function(f)#
Return the function field element coerced from
x.EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(GF(4), 2) # optional - sage.rings.finite_rings sage: C = Curve(x^5 + y^5 + x*y*z^3 + z^5) # optional - sage.rings.finite_rings sage: f = C.function(x/y); f # optional - sage.rings.finite_rings 1/y sage: f.divisor() # optional - sage.rings.finite_rings Place (1/y, 1/y^2*z^2 + z2/y*z + 1) + Place (1/y, 1/y^2*z^2 + ((z2 + 1)/y)*z + 1) + Place (1/y, 1/y*z + 1) - Place (y, z^2 + z2*z + 1) - Place (y, z^2 + (z2 + 1)*z + 1) - Place (y, z + 1)
- function_field()#
Return the function field of this curve.
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve(x^2 + y^2 + z^2, P) sage: C.function_field() Function field in z defined by z^2 + y^2 + 1
sage: P.<x,y,z> = ProjectiveSpace(GF(4), 2) # optional - sage.rings.finite_rings sage: C = Curve(x^5 + y^5 + x*y*z^3 + z^5) # optional - sage.rings.finite_rings sage: C.function_field() # optional - sage.rings.finite_rings Function field in z defined by z^5 + y*z^3 + y^5 + 1
- place_to_closed_point(place)#
Return the closed point at the place.
INPUT:
place– a place of the function field of the curve
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(GF(5), 2) # optional - sage.rings.finite_rings sage: C = Curve(y^2*z^7 - x^9 - x*z^8) # optional - sage.rings.finite_rings sage: pls = C.places() # optional - sage.rings.finite_rings sage: C.place_to_closed_point(pls[-1]) # optional - sage.rings.finite_rings Point (x - 2*z, y - 2*z) sage: pls2 = C.places(2) # optional - sage.rings.finite_rings sage: C.place_to_closed_point(pls2[0]) # optional - sage.rings.finite_rings Point (y^2 + y*z + z^2, x + y)
- places_on(point)#
Return the places on the closed point.
INPUT:
point– a closed point of the curve
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve(x*y*z^4 - x^6 - y^6) sage: C.singular_closed_points() [Point (x, y)] sage: p, = _ sage: C.places_on(p) [Place (1/y, 1/y^2*z, 1/y^3*z^2, 1/y^4*z^3), Place (y, y*z, y*z^2, y*z^3)] sage: pl1, pl2 =_ sage: C.place_to_closed_point(pl1) Point (x, y) sage: C.place_to_closed_point(pl2) Point (x, y)
sage: P.<x,y,z> = ProjectiveSpace(GF(5), 2) # optional - sage.rings.finite_rings sage: C = Curve(x^2*z - y^3) # optional - sage.rings.finite_rings sage: [C.places_on(p) for p in C.closed_points()] # optional - sage.rings.finite_rings [[Place (1/y)], [Place (y)], [Place (y + 1)], [Place (y + 2)], [Place (y + 3)], [Place (y + 4)]]
- singular_closed_points()#
Return the singular closed points of the curve.
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve(y^2*z - x^3, P) sage: C.singular_closed_points() [Point (x, y)]
sage: P.<x,y,z> = ProjectiveSpace(GF(5), 2) # optional - sage.rings.finite_rings sage: C = Curve(y^2*z^7 - x^9 - x*z^8) # optional - sage.rings.finite_rings sage: C.singular_closed_points() # optional - sage.rings.finite_rings [Point (x, z)]
- class sage.schemes.curves.projective_curve.IntegralProjectiveCurve_finite_field(A, f)#
Bases:
IntegralProjectiveCurveIntegral projective curve over a finite field.
INPUT:
A– an ambient projective spacef– homogeneous polynomials defining the curve
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(GF(5), 2) # optional - sage.rings.finite_rings sage: C = Curve(y^2*z^7 - x^9 - x*z^8) # optional - sage.rings.finite_rings sage: C.function_field() # optional - sage.rings.finite_rings Function field in z defined by z^8 + 4*y^2*z^7 + 1 sage: C.closed_points() # optional - sage.rings.finite_rings [Point (x, z), Point (x, y), Point (x - 2*z, y + 2*z), Point (x + 2*z, y + z), Point (x + 2*z, y - z), Point (x - 2*z, y - 2*z)]
- L_polynomial(name='t')#
Return the L-polynomial of this possibly singular curve.
INPUT:
name– (default:t) name of the variable of the polynomial
EXAMPLES:
sage: A.<x,y> = AffineSpace(GF(3), 2) # optional - sage.rings.finite_rings sage: C = Curve(y^2 - x^5 - x^4 - 2*x^3 - 2*x - 2) # optional - sage.rings.finite_rings sage: Cbar = C.projective_closure() # optional - sage.rings.finite_rings sage: Cbar.L_polynomial() # optional - sage.rings.finite_rings 9*t^4 - 3*t^3 + t^2 - t + 1
- closed_points(degree=1)#
Return a list of closed points of
degreeof the curve.INPUT:
degree– a positive integer
EXAMPLES:
sage: A.<x,y> = AffineSpace(GF(9),2) # optional - sage.rings.finite_rings sage: C = Curve(y^2 - x^5 - x^4 - 2*x^3 - 2*x-2) # optional - sage.rings.finite_rings sage: Cp = C.projective_closure() # optional - sage.rings.finite_rings sage: Cp.closed_points() # optional - sage.rings.finite_rings [Point (x0, x1), Point (x0 + (-z2 - 1)*x2, x1), Point (x0 + (z2 + 1)*x2, x1), Point (x0 + z2*x2, x1 + (z2 - 1)*x2), Point (x0 + (-z2)*x2, x1 + (-z2 + 1)*x2), Point (x0 + (-z2 - 1)*x2, x1 + (-z2 - 1)*x2), Point (x0 + (z2 + 1)*x2, x1 + (z2 + 1)*x2), Point (x0 + (z2 - 1)*x2, x1 + z2*x2), Point (x0 + (-z2 + 1)*x2, x1 + (-z2)*x2), Point (x0 + x2, x1 - x2), Point (x0 - x2, x1 + x2)]
- number_of_rational_points(r=1)#
Return the number of rational points of the curve with constant field extended by degree
r.INPUT:
r– positive integer (default: \(1\))
EXAMPLES:
sage: A.<x,y> = AffineSpace(GF(3), 2) # optional - sage.rings.finite_rings sage: C = Curve(y^2 - x^5 - x^4 - 2*x^3 - 2*x - 2) # optional - sage.rings.finite_rings sage: Cbar = C.projective_closure() # optional - sage.rings.finite_rings sage: Cbar.number_of_rational_points(3) # optional - sage.rings.finite_rings 21 sage: D = Cbar.change_ring(Cbar.base_ring().extension(3)) # optional - sage.rings.finite_rings sage: D.base_ring() # optional - sage.rings.finite_rings Finite Field in z3 of size 3^3 sage: len(D.closed_points()) # optional - sage.rings.finite_rings 21
- places(degree=1)#
Return all places on the curve of the
degree.INPUT:
degree– positive integer
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(GF(5), 2) # optional - sage.rings.finite_rings sage: C = Curve(x^2*z - y^3) # optional - sage.rings.finite_rings sage: C.places() # optional - sage.rings.finite_rings [Place (1/y), Place (y), Place (y + 1), Place (y + 2), Place (y + 3), Place (y + 4)] sage: C.places(2) # optional - sage.rings.finite_rings [Place (y^2 + 2), Place (y^2 + 3), Place (y^2 + y + 1), Place (y^2 + y + 2), Place (y^2 + 2*y + 3), Place (y^2 + 2*y + 4), Place (y^2 + 3*y + 3), Place (y^2 + 3*y + 4), Place (y^2 + 4*y + 1), Place (y^2 + 4*y + 2)]
- class sage.schemes.curves.projective_curve.IntegralProjectivePlaneCurve(A, f)#
- class sage.schemes.curves.projective_curve.IntegralProjectivePlaneCurve_finite_field(A, f)#
Bases:
IntegralProjectiveCurve_finite_field,ProjectivePlaneCurve_finite_fieldIntegral projective plane curve over a finite field.
INPUT:
A– ambient projective planef– a homogeneous equation that defines the curve
EXAMPLES:
sage: A.<x,y> = AffineSpace(GF(9), 2) # optional - sage.rings.finite_rings sage: C = Curve(y^2 - x^5 - x^4 - 2*x^3 - 2*x - 2) # optional - sage.rings.finite_rings sage: Cb = C.projective_closure() # optional - sage.rings.finite_rings sage: Cb.singular_closed_points() # optional - sage.rings.finite_rings [Point (x0, x1)] sage: Cb.function_field() # optional - sage.rings.finite_rings Function field in y defined by y^2 + 2*x^5 + 2*x^4 + x^3 + x + 1
- class sage.schemes.curves.projective_curve.ProjectiveCurve(A, X)#
Bases:
Curve_generic,AlgebraicScheme_subscheme_projectiveCurves in projective spaces.
INPUT:
A– ambient projective spaceX– list of multivariate polynomials; defining equations of the curve
EXAMPLES:
sage: P.<x,y,z,w,u> = ProjectiveSpace(GF(7), 4) # optional - sage.rings.finite_rings sage: C = Curve([y*u^2 - x^3, z*u^2 - x^3, w*u^2 - x^3, y^3 - x^3], P); C # optional - sage.rings.finite_rings Projective Curve over Finite Field of size 7 defined by -x^3 + y*u^2, -x^3 + z*u^2, -x^3 + w*u^2, -x^3 + y^3
sage: K.<u> = CyclotomicField(11) # optional - sage.rings.number_field sage: P.<x,y,z,w> = ProjectiveSpace(K, 3) # optional - sage.rings.number_field sage: C = Curve([y*w - u*z^2 - x^2, x*w - 3*u^2*z*w], P); C # optional - sage.rings.number_field Projective Curve over Cyclotomic Field of order 11 and degree 10 defined by -x^2 + (-u)*z^2 + y*w, x*w + (-3*u^2)*z*w
- affine_patch(i, AA=None)#
Return the \(i\)-th affine patch of this projective curve.
INPUT:
i– affine coordinate chart of the projective ambient space of this curve to compute affine patch with respect toAA– (default: None) ambient affine space, this is constructed if it is not given
OUTPUT: A curve in affine space.
EXAMPLES:
sage: P.<x,y,z,w> = ProjectiveSpace(CC, 3) sage: C = Curve([y*z - x^2, w^2 - x*y], P) sage: C.affine_patch(0) Affine Curve over Complex Field with 53 bits of precision defined by y*z - 1.00000000000000, w^2 - y
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve(x^3 - x^2*y + y^3 - x^2*z, P) sage: C.affine_patch(1) Affine Plane Curve over Rational Field defined by x^3 - x^2*z - x^2 + 1
sage: A.<x,y> = AffineSpace(QQ, 2) sage: P.<u,v,w> = ProjectiveSpace(QQ, 2) sage: C = Curve([u^2 - v^2], P) sage: C.affine_patch(1, A).ambient_space() == A True
- plane_projection(PP=None)#
Return a projection of this curve into a projective plane.
INPUT:
PP– (default: None) the projective plane the projected curve will be defined in. This space must be defined over the same base field as this curve, and must have dimension two. This space is constructed if not specified.
OUTPUT: A tuple of
a scheme morphism from this curve into a projective plane
the projective curve that is the image of that morphism
EXAMPLES:
sage: P.<x,y,z,w,u,v> = ProjectiveSpace(QQ, 5) sage: C = P.curve([x*u - z*v, w - y, w*y - x^2, y^3*u*2*z - w^4*w]) sage: L.<a,b,c> = ProjectiveSpace(QQ, 2) sage: proj1 = C.plane_projection(PP=L) sage: proj1 (Scheme morphism: From: Projective Curve over Rational Field defined by x*u - z*v, -y + w, -x^2 + y*w, -w^5 + 2*y^3*z*u To: Projective Space of dimension 2 over Rational Field Defn: Defined on coordinates by sending (x : y : z : w : u : v) to (x : -z + u : -z + v), Projective Plane Curve over Rational Field defined by a^8 + 6*a^7*b + 4*a^5*b^3 - 4*a^7*c - 2*a^6*b*c - 4*a^5*b^2*c + 2*a^6*c^2) sage: proj1[1].ambient_space() is L True sage: proj2 = C.projection() sage: proj2[1].ambient_space() is L False
sage: P.<x,y,z,w,u> = ProjectiveSpace(GF(7), 4) # optional - sage.rings.finite_rings sage: C = P.curve([x^2 - 6*y^2, w*z*u - y^3 + 4*y^2*z, u^2 - x^2]) # optional - sage.rings.finite_rings sage: C.plane_projection() # optional - sage.rings.finite_rings (Scheme morphism: From: Projective Curve over Finite Field of size 7 defined by x^2 + y^2, -y^3 - 3*y^2*z + z*w*u, -x^2 + u^2 To: Projective Space of dimension 2 over Finite Field of size 7 Defn: Defined on coordinates by sending (x : y : z : w : u) to (x : z : -y + w), Projective Plane Curve over Finite Field of size 7 defined by x0^10 + 2*x0^8*x1^2 + 2*x0^6*x1^4 - 3*x0^6*x1^3*x2 + 2*x0^6*x1^2*x2^2 - 2*x0^4*x1^4*x2^2 + x0^2*x1^4*x2^4)
sage: P.<x,y,z> = ProjectiveSpace(GF(17), 2) # optional - sage.rings.finite_rings sage: C = P.curve(x^2 - y*z - z^2) # optional - sage.rings.finite_rings sage: C.plane_projection() # optional - sage.rings.finite_rings Traceback (most recent call last): ... TypeError: this curve is already a plane curve
- projection(P=None, PS=None)#
Return a projection of this curve into projective space of dimension one less than the dimension of the ambient space of this curve.
This curve must not already be a plane curve. Over finite fields, if this curve contains all points in its ambient space, then an error will be returned.
INPUT:
P– (default: None) a point not on this curve that will be used to define the projection map; this is constructed if not specified.PS– (default: None) the projective space the projected curve will be defined in. This space must be defined over the same base ring as this curve, and must have dimension one less than that of the ambient space of this curve. This space will be constructed if not specified.
OUTPUT: A tuple of
a scheme morphism from this curve into a projective space of dimension one less than that of the ambient space of this curve
the projective curve that is the image of that morphism
EXAMPLES:
sage: K.<a> = CyclotomicField(3) # optional - sage.rings.number_field sage: P.<x,y,z,w> = ProjectiveSpace(K, 3) # optional - sage.rings.number_field sage: C = Curve([y*w - x^2, z*w^2 - a*x^3], P) # optional - sage.rings.number_field sage: L.<a,b,c> = ProjectiveSpace(K, 2) # optional - sage.rings.number_field sage: proj1 = C.projection(PS=L) # optional - sage.rings.number_field sage: proj1 # optional - sage.rings.number_field (Scheme morphism: From: Projective Curve over Cyclotomic Field of order 3 and degree 2 defined by -x^2 + y*w, (-a)*x^3 + z*w^2 To: Projective Space of dimension 2 over Cyclotomic Field of order 3 and degree 2 Defn: Defined on coordinates by sending (x : y : z : w) to (x : y : -z + w), Projective Plane Curve over Cyclotomic Field of order 3 and degree 2 defined by a^6 + (-a)*a^3*b^3 - a^4*b*c) sage: proj1[1].ambient_space() is L True sage: proj2 = C.projection() sage: proj2[1].ambient_space() is L False
sage: P.<x,y,z,w,a,b,c> = ProjectiveSpace(QQ, 6) sage: C = Curve([y - x, z - a - b, w^2 - c^2, z - x - a, x^2 - w*z], P) sage: C.projection() (Scheme morphism: From: Projective Curve over Rational Field defined by -x + y, z - a - b, w^2 - c^2, -x + z - a, x^2 - z*w To: Projective Space of dimension 5 over Rational Field Defn: Defined on coordinates by sending (x : y : z : w : a : b : c) to (x : y : -z + w : a : b : c), Projective Curve over Rational Field defined by x1 - x4, x0 - x4, x2*x3 + x3^2 + x2*x4 + 2*x3*x4, x2^2 - x3^2 - 2*x3*x4 + x4^2 - x5^2, x2*x4^2 + x3*x4^2 + x4^3 - x3*x5^2 - x4*x5^2, x4^4 - x3^2*x5^2 - 2*x3*x4*x5^2 - x4^2*x5^2)
sage: P.<x,y,z,w> = ProjectiveSpace(GF(2), 3) # optional - sage.rings.finite_rings sage: C = P.curve([(x - y)*(x - z)*(x - w)*(y - z)*(y - w), # optional - sage.rings.finite_rings ....: x*y*z*w*(x + y + z + w)]) sage: C.projection() # optional - sage.rings.finite_rings Traceback (most recent call last): ... NotImplementedError: this curve contains all points of its ambient space
sage: P.<x,y,z,w,u> = ProjectiveSpace(GF(7), 4) # optional - sage.rings.finite_rings sage: C = P.curve([x^3 - y*z*u, w^2 - u^2 + 2*x*z, 3*x*w - y^2]) # optional - sage.rings.finite_rings sage: L.<a,b,c,d> = ProjectiveSpace(GF(7), 3) # optional - sage.rings.finite_rings sage: C.projection(PS=L) # optional - sage.rings.finite_rings (Scheme morphism: From: Projective Curve over Finite Field of size 7 defined by x^3 - y*z*u, 2*x*z + w^2 - u^2, -y^2 + 3*x*w To: Projective Space of dimension 3 over Finite Field of size 7 Defn: Defined on coordinates by sending (x : y : z : w : u) to (x : y : z : w), Projective Curve over Finite Field of size 7 defined by b^2 - 3*a*d, a^5*b + a*b*c^3*d - 3*b*c^2*d^3, a^6 + a^2*c^3*d - 3*a*c^2*d^3) sage: Q.<a,b,c> = ProjectiveSpace(GF(7), 2) # optional - sage.rings.finite_rings sage: C.projection(PS=Q) # optional - sage.rings.finite_rings Traceback (most recent call last): ... TypeError: (=Projective Space of dimension 2 over Finite Field of size 7) must have dimension (=3)
sage: PP.<x,y,z,w> = ProjectiveSpace(QQ, 3) sage: C = PP.curve([x^3 - z^2*y, w^2 - z*x]) sage: Q = PP([1,0,1,1]) sage: C.projection(P=Q) (Scheme morphism: From: Projective Curve over Rational Field defined by x^3 - y*z^2, -x*z + w^2 To: Projective Space of dimension 2 over Rational Field Defn: Defined on coordinates by sending (x : y : z : w) to (y : -x + z : -x + w), Projective Plane Curve over Rational Field defined by x0*x1^5 - 6*x0*x1^4*x2 + 14*x0*x1^3*x2^2 - 16*x0*x1^2*x2^3 + 9*x0*x1*x2^4 - 2*x0*x2^5 - x2^6) sage: LL.<a,b,c> = ProjectiveSpace(QQ, 2) sage: Q = PP([0,0,0,1]) sage: C.projection(PS=LL, P=Q) (Scheme morphism: From: Projective Curve over Rational Field defined by x^3 - y*z^2, -x*z + w^2 To: Projective Space of dimension 2 over Rational Field Defn: Defined on coordinates by sending (x : y : z : w) to (x : y : z), Projective Plane Curve over Rational Field defined by a^3 - b*c^2) sage: Q = PP([0,0,1,0]) sage: C.projection(P=Q) Traceback (most recent call last): ... TypeError: (=(0 : 0 : 1 : 0)) must be a point not on this curve
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = P.curve(y^2 - x^2 + z^2) sage: C.projection() Traceback (most recent call last): ... TypeError: this curve is already a plane curve
- class sage.schemes.curves.projective_curve.ProjectiveCurve_field(A, X)#
Bases:
ProjectiveCurve,AlgebraicScheme_subscheme_projective_fieldProjective curves over fields.
- arithmetic_genus()#
Return the arithmetic genus of this projective curve.
This is the arithmetic genus \(g_a(C)\) as defined in [Har1977]. If \(P\) is the Hilbert polynomial of the defining ideal of this curve, then the arithmetic genus of this curve is \(1 - P(0)\). This curve must be irreducible.
EXAMPLES:
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3) sage: C = P.curve([w*z - x^2, w^2 + y^2 + z^2]) sage: C.arithmetic_genus() 1
sage: P.<x,y,z,w,t> = ProjectiveSpace(GF(7), 4) # optional - sage.rings.finite_rings sage: C = P.curve([t^3 - x*y*w, x^3 + y^3 + z^3, z - w]) # optional - sage.rings.finite_rings sage: C.arithmetic_genus() # optional - sage.rings.finite_rings 10
- is_complete_intersection()#
Return whether this projective curve is a complete intersection.
EXAMPLES:
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3) sage: C = Curve([x*y - z*w, x^2 - y*w, y^2*w - x*z*w], P) sage: C.is_complete_intersection() False
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3) sage: C = Curve([y*w - x^2, z*w^2 - x^3], P) sage: C.is_complete_intersection() True sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3) sage: C = Curve([z^2 - y*w, y*z - x*w, y^2 - x*z], P) sage: C.is_complete_intersection() False
- tangent_line(p)#
Return the tangent line at the point
p.INPUT:
p– a rational point of the curve
EXAMPLES:
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3) sage: C = Curve([x*y - z*w, x^2 - y*w, y^2*w - x*z*w], P) sage: p = C(1,1,1,1) sage: C.tangent_line(p) Projective Curve over Rational Field defined by -2*x + y + w, -3*x + z + 2*w
- class sage.schemes.curves.projective_curve.ProjectivePlaneCurve(A, f)#
Bases:
ProjectiveCurveCurves in projective planes.
INPUT:
A– projective planef– homogeneous polynomial in the homogeneous coordinate ring of the plane
EXAMPLES:
A projective plane curve defined over an algebraic closure of \(\QQ\):
sage: P.<x,y,z> = ProjectiveSpace(QQbar, 2) # optional - sage.rings.number_field sage: set_verbose(-1) # suppress warnings for slow computation # optional - sage.rings.number_field sage: C = Curve([y*z - x^2 - QQbar.gen()*z^2], P); C # optional - sage.rings.number_field Projective Plane Curve over Algebraic Field defined by -x^2 + y*z + (-I)*z^2
A projective plane curve defined over a finite field:
sage: P.<x,y,z> = ProjectiveSpace(GF(5^2, 'v'), 2) # optional - sage.rings.finite_rings sage: C = Curve([y^2*z - x*z^2 - z^3], P); C # optional - sage.rings.finite_rings Projective Plane Curve over Finite Field in v of size 5^2 defined by y^2*z - x*z^2 - z^3
- degree()#
Return the degree of this projective curve.
For a plane curve, this is just the degree of its defining polynomial.
OUTPUT: An integer.
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = P.curve([y^7 - x^2*z^5 + 7*z^7]) sage: C.degree() 7
- divisor_of_function(r)#
Return the divisor of a function on a curve.
INPUT:
ris a rational function on XOUTPUT: A 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: FF = FiniteField(5) # optional - sage.rings.finite_rings sage: P2 = ProjectiveSpace(2, FF, names=['x','y','z']) # optional - sage.rings.finite_rings sage: R = P2.coordinate_ring() # optional - sage.rings.finite_rings sage: x, y, z = R.gens() # optional - sage.rings.finite_rings sage: f = y^2*z^7 - x^9 - x*z^8 # optional - sage.rings.finite_rings sage: C = Curve(f) # optional - sage.rings.finite_rings sage: K = FractionField(R) # optional - sage.rings.finite_rings sage: r = 1/x # optional - sage.rings.finite_rings sage: C.divisor_of_function(r) # todo: not implemented !!!! # optional - sage.rings.finite_rings [[-1, (0, 0, 1)]] sage: r = 1/x^3 # optional - sage.rings.finite_rings sage: C.divisor_of_function(r) # todo: not implemented !!!! # optional - sage.rings.finite_rings [[-3, (0, 0, 1)]]
- excellent_position(Q)#
Return a transformation of this curve into one in excellent position with respect to the point
Q.Here excellent position is defined as in [Ful1989]. A curve \(C\) of degree \(d\) containing the point \((0 : 0 : 1)\) with multiplicity \(r\) is said to be in excellent position if none of the coordinate lines are tangent to \(C\) at any of the fundamental points \((1 : 0 : 0)\), \((0 : 1 : 0)\), and \((0 : 0 : 1)\), and if the two coordinate lines containing \((0 : 0 : 1)\) intersect \(C\) transversally in \(d - r\) distinct non-fundamental points, and if the other coordinate line intersects \(C\) transversally at \(d\) distinct, non-fundamental points.
INPUT:
Q– a point on this curve.
OUTPUT:
a scheme morphism from this curve to a curve in excellent position that is a restriction of a change of coordinates map of the projective plane.
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([x*y - z^2], P) sage: Q = P([1,1,1]) sage: C.excellent_position(Q) Scheme morphism: From: Projective Plane Curve over Rational Field defined by x*y - z^2 To: Projective Plane Curve over Rational Field defined by -x^2 - 3*x*y - 4*y^2 - x*z - 3*y*z Defn: Defined on coordinates by sending (x : y : z) to (-x + 1/2*y + 1/2*z : -1/2*y + 1/2*z : x + 1/2*y - 1/2*z)
sage: R.<a> = QQ[] sage: K.<b> = NumberField(a^2 - 3) # optional - sage.rings.number_field sage: P.<x,y,z> = ProjectiveSpace(K, 2) # optional - sage.rings.number_field sage: C = P.curve([z^2*y^3*x^4 - y^6*x^3 - 4*z^2*y^4*x^3 - 4*z^4*y^2*x^3 # optional - sage.rings.number_field ....: + 3*y^7*x^2 + 10*z^2*y^5*x^2 + 9*z^4*y^3*x^2 ....: + 5*z^6*y*x^2 - 3*y^8*x - 9*z^2*y^6*x - 11*z^4*y^4*x ....: - 7*z^6*y^2*x - 2*z^8*x + y^9 + 2*z^2*y^7 + 3*z^4*y^5 ....: + 4*z^6*y^3 + 2*z^8*y]) sage: Q = P([1,0,0]) # optional - sage.rings.number_field sage: C.excellent_position(Q) # optional - sage.rings.number_field Scheme morphism: From: Projective Plane Curve over Number Field in b with defining polynomial a^2 - 3 defined by -x^3*y^6 + 3*x^2*y^7 - 3*x*y^8 + y^9 + x^4*y^3*z^2 - 4*x^3*y^4*z^2 + 10*x^2*y^5*z^2 - 9*x*y^6*z^2 + 2*y^7*z^2 - 4*x^3*y^2*z^4 + 9*x^2*y^3*z^4 - 11*x*y^4*z^4 + 3*y^5*z^4 + 5*x^2*y*z^6 - 7*x*y^2*z^6 + 4*y^3*z^6 - 2*x*z^8 + 2*y*z^8 To: Projective Plane Curve over Number Field in b with defining polynomial a^2 - 3 defined by 900*x^9 - 7410*x^8*y + 29282*x^7*y^2 - 69710*x^6*y^3 + 110818*x^5*y^4 - 123178*x^4*y^5 + 96550*x^3*y^6 - 52570*x^2*y^7 + 18194*x*y^8 - 3388*y^9 - 1550*x^8*z + 9892*x^7*y*z - 30756*x^6*y^2*z + 58692*x^5*y^3*z - 75600*x^4*y^4*z + 67916*x^3*y^5*z - 42364*x^2*y^6*z + 16844*x*y^7*z - 3586*y^8*z + 786*x^7*z^2 - 3958*x^6*y*z^2 + 9746*x^5*y^2*z^2 - 14694*x^4*y^3*z^2 + 15174*x^3*y^4*z^2 - 10802*x^2*y^5*z^2 + 5014*x*y^6*z^2 - 1266*y^7*z^2 - 144*x^6*z^3 + 512*x^5*y*z^3 - 912*x^4*y^2*z^3 + 1024*x^3*y^3*z^3 - 816*x^2*y^4*z^3 + 512*x*y^5*z^3 - 176*y^6*z^3 + 8*x^5*z^4 - 8*x^4*y*z^4 - 16*x^3*y^2*z^4 + 16*x^2*y^3*z^4 + 8*x*y^4*z^4 - 8*y^5*z^4 Defn: Defined on coordinates by sending (x : y : z) to (1/4*y + 1/2*z : -1/4*y + 1/2*z : x + 1/4*y - 1/2*z)
sage: set_verbose(-1) sage: a = QQbar(sqrt(2)) # optional - sage.rings.number_field sage: P.<x,y,z> = ProjectiveSpace(QQbar, 2) # optional - sage.rings.number_field sage: C = Curve([(-1/4*a)*x^3 + (-3/4*a)*x^2*y # optional - sage.rings.number_field ....: + (-3/4*a)*x*y^2 + (-1/4*a)*y^3 - 2*x*y*z], P) sage: Q = P([0,0,1]) # optional - sage.rings.number_field sage: C.excellent_position(Q) # optional - sage.rings.number_field Scheme morphism: From: Projective Plane Curve over Algebraic Field defined by (-0.3535533905932738?)*x^3 + (-1.060660171779822?)*x^2*y + (-1.060660171779822?)*x*y^2 + (-0.3535533905932738?)*y^3 + (-2)*x*y*z To: Projective Plane Curve over Algebraic Field defined by (-2.828427124746190?)*x^3 + (-2)*x^2*y + 2*y^3 + (-2)*x^2*z + 2*y^2*z Defn: Defined on coordinates by sending (x : y : z) to (1/2*x + 1/2*y : (-1/2)*x + 1/2*y : 1/2*x + (-1/2)*y + z)
- is_ordinary_singularity(P)#
Return whether the singular point
Pof this projective plane curve is an ordinary singularity.The point
Pis an ordinary singularity of this curve if it is a singular point, and if the tangents of this curve atPare distinct.INPUT:
P– a point on this curve.
OUTPUT:
Boolean. True or False depending on whether
Pis or is not an ordinary singularity of this curve, respectively. An error is raised ifPis not a singular point of this curve.
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([y^2*z^3 - x^5], P) sage: Q = P([0,0,1]) sage: C.is_ordinary_singularity(Q) False
sage: R.<a> = QQ[] sage: K.<b> = NumberField(a^2 - 3) # optional - sage.rings.number_field sage: P.<x,y,z> = ProjectiveSpace(K, 2) # optional - sage.rings.number_field sage: C = P.curve([x^2*y^3*z^4 - y^6*z^3 - 4*x^2*y^4*z^3 - 4*x^4*y^2*z^3 # optional - sage.rings.number_field ....: + 3*y^7*z^2 + 10*x^2*y^5*z^2 + 9*x^4*y^3*z^2 ....: + 5*x^6*y*z^2 - 3*y^8*z - 9*x^2*y^6*z - 11*x^4*y^4*z ....: - 7*x^6*y^2*z - 2*x^8*z + y^9 + 2*x^2*y^7 + 3*x^4*y^5 ....: + 4*x^6*y^3 + 2*x^8*y]) sage: Q = P([0,1,1]) # optional - sage.rings.number_field sage: C.is_ordinary_singularity(Q) # optional - sage.rings.number_field True
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = P.curve([z^5 - y^5 + x^5 + x*y^2*z^2]) sage: Q = P([0,1,1]) sage: C.is_ordinary_singularity(Q) Traceback (most recent call last): ... TypeError: (=(0 : 1 : 1)) is not a singular point of (=Projective Plane Curve over Rational Field defined by x^5 - y^5 + x*y^2*z^2 + z^5)
- is_singular(P=None)#
Return whether this curve is singular or not, or if a point
Pis provided, whetherPis a singular point of this curve.INPUT:
P– (default:None) a point on this curve
OUTPUT:
If no point
Pis provided, returnTrueorFalsedepending on whether this curve is singular or not. If a pointPis provided, returnTrueorFalsedepending on whetherPis or is not a singular point of this curve.EXAMPLES:
Over \(\QQ\):
sage: F = QQ sage: P2.<X,Y,Z> = ProjectiveSpace(F, 2) sage: C = Curve(X^3 - Y^2*Z) sage: C.is_singular() True
Over a finite field:
sage: F = GF(19) # optional - sage.rings.finite_rings sage: P2.<X,Y,Z> = ProjectiveSpace(F, 2) # optional - sage.rings.finite_rings sage: C = Curve(X^3 + Y^3 + Z^3) # optional - sage.rings.finite_rings sage: C.is_singular() # optional - sage.rings.finite_rings False sage: D = Curve(X^4 - X*Z^3) # optional - sage.rings.finite_rings sage: D.is_singular() # optional - sage.rings.finite_rings True sage: E = Curve(X^5 + 19*Y^5 + Z^5) # optional - sage.rings.finite_rings sage: E.is_singular() # optional - sage.rings.finite_rings True sage: E = Curve(X^5 + 9*Y^5 + Z^5) # optional - sage.rings.finite_rings sage: E.is_singular() # optional - sage.rings.finite_rings False
Over \(\CC\):
sage: F = CC sage: P2.<X,Y,Z> = ProjectiveSpace(F, 2) sage: C = Curve(X) sage: C.is_singular() False sage: D = Curve(Y^2*Z - X^3) sage: D.is_singular() True sage: E = Curve(Y^2*Z - X^3 + Z^3) sage: E.is_singular() False
Showing that github issue #12187 is fixed:
sage: F.<X,Y,Z> = GF(2)[] # optional - sage.rings.finite_rings sage: G = Curve(X^2 + Y*Z) # optional - sage.rings.finite_rings sage: G.is_singular() # optional - sage.rings.finite_rings False
sage: P.<x,y,z> = ProjectiveSpace(CC, 2) sage: C = Curve([y^4 - x^3*z], P) sage: Q = P([0,0,1]) sage: C.is_singular() True
- is_transverse(C, P)#
Return whether the intersection of this curve with the curve
Cat the pointPis transverse.The intersection at
Pis transverse ifPis a nonsingular point of both curves, and if the tangents of the curves atPare distinct.INPUT:
C– a curve in the ambient space of this curve.P– a point in the intersection of both curves.
OUTPUT: A boolean.
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([x^2 - y^2], P) sage: D = Curve([x - y], P) sage: Q = P([1,1,0]) sage: C.is_transverse(D, Q) False
sage: K = QuadraticField(-1) # optional - sage.rings.number_field sage: P.<x,y,z> = ProjectiveSpace(K, 2) # optional - sage.rings.number_field sage: C = Curve([y^2*z - K.0*x^3], P) # optional - sage.rings.number_field sage: D = Curve([z*x + y^2], P) # optional - sage.rings.number_field sage: Q = P([0,0,1]) # optional - sage.rings.number_field sage: C.is_transverse(D, Q) # optional - sage.rings.number_field False
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([x^2 - 2*y^2 - 2*z^2], P) sage: D = Curve([y - z], P) sage: Q = P([2,1,1]) 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 worse than others.
INPUT:
pt– a rational point on X which is not a point of ramificationfor the projection \((x,y) \to x\).
n– the number of terms desired
OUTPUT: \(x = x0 + t\), \(y = y0\) + power series in \(t\)
EXAMPLES:
sage: FF = FiniteField(5) # optional - sage.rings.finite_rings sage: P2 = ProjectiveSpace(2, FF, names=['x','y','z']) # optional - sage.rings.finite_rings sage: x, y, z = P2.coordinate_ring().gens() # optional - sage.rings.finite_rings sage: C = Curve(y^2*z^7 - x^9 - x*z^8) # optional - sage.rings.finite_rings sage: pt = C([2,3,1]) # optional - sage.rings.finite_rings sage: C.local_coordinates(pt,9) # todo: not implemented !!!! # optional - sage.rings.finite_rings [2 + t, 3 + 3*t^2 + t^3 + 3*t^4 + 3*t^6 + 3*t^7 + t^8 + 2*t^9 + 3*t^11 + 3*t^12]
- ordinary_model()#
Return a birational map from this curve to a plane curve with only ordinary singularities.
Currently only implemented over number fields. If not all of the coordinates of the non-ordinary singularities of this curve are contained in its base field, then the domain and codomain of the map returned will be defined over an extension. This curve must be irreducible.
OUTPUT:
a scheme morphism from this curve to a curve with only ordinary singularities that defines a birational map between the two curves.
EXAMPLES:
sage: set_verbose(-1) sage: K = QuadraticField(3) # optional - sage.rings.number_field sage: P.<x,y,z> = ProjectiveSpace(K, 2) # optional - sage.rings.number_field sage: C = Curve([x^5 - K.0*y*z^4], P) # optional - sage.rings.number_field sage: C.ordinary_model() # optional - sage.rings.number_field Scheme morphism: From: Projective Plane Curve over Number Field in a with defining polynomial x^2 - 3 with a = 1.732050807568878? defined by x^5 + (-a)*y*z^4 To: Projective Plane Curve over Number Field in a with defining polynomial x^2 - 3 with a = 1.732050807568878? defined by (-a)*x^5*y + (-4*a)*x^4*y^2 + (-6*a)*x^3*y^3 + (-4*a)*x^2*y^4 + (-a)*x*y^5 + (-a - 1)*x^5*z + (-4*a + 5)*x^4*y*z + (-6*a - 10)*x^3*y^2*z + (-4*a + 10)*x^2*y^3*z + (-a - 5)*x*y^4*z + y^5*z Defn: Defined on coordinates by sending (x : y : z) to (-1/4*x^2 - 1/2*x*y + 1/2*x*z + 1/2*y*z - 1/4*z^2 : 1/4*x^2 + 1/2*x*y + 1/2*y*z - 1/4*z^2 : -1/4*x^2 + 1/4*z^2)
sage: set_verbose(-1) sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([y^2*z^2 - x^4 - x^3*z], P) sage: D = C.ordinary_model(); D # long time (2 seconds) Scheme morphism: From: Projective Plane Curve over Rational Field defined by -x^4 - x^3*z + y^2*z^2 To: Projective Plane Curve over Rational Field defined by 4*x^6*y^3 - 24*x^5*y^4 + 36*x^4*y^5 + 8*x^6*y^2*z - 40*x^5*y^3*z + 24*x^4*y^4*z + 72*x^3*y^5*z - 4*x^6*y*z^2 + 8*x^5*y^2*z^2 - 56*x^4*y^3*z^2 + 104*x^3*y^4*z^2 + 44*x^2*y^5*z^2 + 8*x^6*z^3 - 16*x^5*y*z^3 - 24*x^4*y^2*z^3 + 40*x^3*y^3*z^3 + 48*x^2*y^4*z^3 + 8*x*y^5*z^3 - 8*x^5*z^4 + 36*x^4*y*z^4 - 56*x^3*y^2*z^4 + 20*x^2*y^3*z^4 + 40*x*y^4*z^4 - 16*y^5*z^4 Defn: Defined on coordinates by sending (x : y : z) to (-3/64*x^4 + 9/64*x^2*y^2 - 3/32*x*y^3 - 1/16*x^3*z - 1/8*x^2*y*z + 1/4*x*y^2*z - 1/16*y^3*z - 1/8*x*y*z^2 + 1/16*y^2*z^2 : -1/64*x^4 + 3/64*x^2*y^2 - 1/32*x*y^3 + 1/16*x*y^2*z - 1/16*y^3*z + 1/16*y^2*z^2 : 3/64*x^4 - 3/32*x^3*y + 3/64*x^2*y^2 + 1/16*x^3*z - 3/16*x^2*y*z + 1/8*x*y^2*z - 1/8*x*y*z^2 + 1/16*y^2*z^2) sage: all(D.codomain().is_ordinary_singularity(Q) # long time ....: for Q in D.codomain().singular_points()) True
sage: set_verbose(-1) sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([(x^2 + y^2 - y*z - 2*z^2)*(y*z - x^2 + 2*z^2)*z + y^5], P) sage: C.ordinary_model() # long time (5 seconds) Scheme morphism: From: Projective Plane Curve over Number Field in a with defining polynomial y^2 - 2 defined by y^5 - x^4*z - x^2*y^2*z + 2*x^2*y*z^2 + y^3*z^2 + 4*x^2*z^3 + y^2*z^3 - 4*y*z^4 - 4*z^5 To: Projective Plane Curve over Number Field in a with defining polynomial y^2 - 2 defined by (-29*a + 1)*x^8*y^6 + (10*a + 158)*x^7*y^7 + (-109*a - 31)*x^6*y^8 + (-80*a - 198)*x^8*y^5*z + (531*a + 272)*x^7*y^6*z + (170*a - 718)*x^6*y^7*z + (19*a - 636)*x^5*y^8*z + (-200*a - 628)*x^8*y^4*z^2 + (1557*a - 114)*x^7*y^5*z^2 + (2197*a - 2449)*x^6*y^6*z^2 + (1223*a - 3800)*x^5*y^7*z^2 + (343*a - 1329)*x^4*y^8*z^2 + (-323*a - 809)*x^8*y^3*z^3 + (1630*a - 631)*x^7*y^4*z^3 + (4190*a - 3126)*x^6*y^5*z^3 + (3904*a - 7110)*x^5*y^6*z^3 + (1789*a - 5161)*x^4*y^7*z^3 + (330*a - 1083)*x^3*y^8*z^3 + (-259*a - 524)*x^8*y^2*z^4 + (720*a - 605)*x^7*y^3*z^4 + (3082*a - 2011)*x^6*y^4*z^4 + (4548*a - 5462)*x^5*y^5*z^4 + (2958*a - 6611)*x^4*y^6*z^4 + (994*a - 2931)*x^3*y^7*z^4 + (117*a - 416)*x^2*y^8*z^4 + (-108*a - 184)*x^8*y*z^5 + (169*a - 168)*x^7*y^2*z^5 + (831*a - 835)*x^6*y^3*z^5 + (2225*a - 1725)*x^5*y^4*z^5 + (1970*a - 3316)*x^4*y^5*z^5 + (952*a - 2442)*x^3*y^6*z^5 + (217*a - 725)*x^2*y^7*z^5 + (16*a - 77)*x*y^8*z^5 + (-23*a - 35)*x^8*z^6 + (43*a + 24)*x^7*y*z^6 + (21*a - 198)*x^6*y^2*z^6 + (377*a - 179)*x^5*y^3*z^6 + (458*a - 537)*x^4*y^4*z^6 + (288*a - 624)*x^3*y^5*z^6 + (100*a - 299)*x^2*y^6*z^6 + (16*a - 67)*x*y^7*z^6 - 5*y^8*z^6 Defn: Defined on coordinates by sending (x : y : z) to ((-5/128*a - 5/128)*x^4 + (-5/32*a + 5/32)*x^3*y + (-1/16*a + 3/32)*x^2*y^2 + (1/16*a - 1/16)*x*y^3 + (1/32*a - 1/32)*y^4 - 1/32*x^3*z + (3/16*a - 5/8)*x^2*y*z + (1/8*a - 5/16)*x*y^2*z + (1/8*a + 5/32)*x^2*z^2 + (-3/16*a + 5/16)*x*y*z^2 + (-3/16*a - 1/16)*y^2*z^2 + 1/16*x*z^3 + (1/4*a + 1/4)*y*z^3 + (-3/32*a - 5/32)*z^4 : (-5/128*a - 5/128)*x^4 + (5/32*a)*x^3*y + (3/32*a + 3/32)*x^2*y^2 + (-1/16*a)*x*y^3 + (-1/32*a - 1/32)*y^4 - 1/32*x^3*z + (-11/32*a)*x^2*y*z + (1/8*a + 5/16)*x*y^2*z + (3/16*a + 1/4)*y^3*z + (1/8*a + 5/32)*x^2*z^2 + (-1/16*a - 3/8)*x*y*z^2 + (-3/8*a - 9/16)*y^2*z^2 + 1/16*x*z^3 + (5/16*a + 1/2)*y*z^3 + (-3/32*a - 5/32)*z^4 : (1/64*a + 3/128)*x^4 + (-1/32*a - 1/32)*x^3*y + (3/32*a - 9/32)*x^2*y^2 + (1/16*a - 3/16)*x*y^3 - 1/32*y^4 + (3/32*a + 1/8)*x^2*y*z + (-1/8*a + 1/8)*x*y^2*z + (-1/16*a)*y^3*z + (-1/16*a - 3/32)*x^2*z^2 + (1/16*a + 1/16)*x*y*z^2 + (3/16*a + 3/16)*y^2*z^2 + (-3/16*a - 1/4)*y*z^3 + (1/16*a + 3/32)*z^4)
- plot(*args, **kwds)#
Plot the real points of an affine patch of this projective plane curve.
INPUT:
self- an affine plane curvepatch- (optional) the affine patch to be plotted; if not specified, the patch corresponding to the last projective coordinate being nonzero*args- optional tuples (variable, minimum, maximum) for plotting dimensions**kwds- optional keyword arguments passed on toimplicit_plot
EXAMPLES:
A cuspidal curve:
sage: R.<x, y, z> = QQ[] sage: C = Curve(x^3 - y^2*z) sage: C.plot() # optional - sage.plot Graphics object consisting of 1 graphics primitive
The other affine patches of the same curve:
sage: C.plot(patch=0) # optional - sage.plot Graphics object consisting of 1 graphics primitive sage: C.plot(patch=1) # optional - sage.plot Graphics object consisting of 1 graphics primitive
An elliptic curve:
sage: E = EllipticCurve('101a') sage: C = Curve(E) sage: C.plot() # optional - sage.plot Graphics object consisting of 1 graphics primitive sage: C.plot(patch=0) # optional - sage.plot Graphics object consisting of 1 graphics primitive sage: C.plot(patch=1) # optional - sage.plot Graphics object consisting of 1 graphics primitive
A hyperelliptic curve:
sage: P.<x> = QQ[] sage: f = 4*x^5 - 30*x^3 + 45*x - 22 sage: C = HyperellipticCurve(f) sage: C.plot() # optional - sage.plot Graphics object consisting of 1 graphics primitive sage: C.plot(patch=0) # optional - sage.plot Graphics object consisting of 1 graphics primitive sage: C.plot(patch=1) # optional - sage.plot Graphics object consisting of 1 graphics primitive
- quadratic_transform()#
Return a birational map from this curve to the proper transform of this curve with respect to the standard Cremona transformation.
The standard Cremona transformation is the birational automorphism of \(\mathbb{P}^{2}\) defined \((x : y : z)\mapsto (yz : xz : xy)\).
OUTPUT:
a scheme morphism representing the restriction of the standard Cremona transformation from this curve to the proper transform.
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve(x^3*y - z^4 - z^2*x^2, P) sage: C.quadratic_transform() Scheme morphism: From: Projective Plane Curve over Rational Field defined by x^3*y - x^2*z^2 - z^4 To: Projective Plane Curve over Rational Field defined by -x^3*y - x*y*z^2 + z^4 Defn: Defined on coordinates by sending (x : y : z) to (y*z : x*z : x*y)
sage: P.<x,y,z> = ProjectiveSpace(GF(17), 2) # optional - sage.rings.finite_rings sage: C = P.curve([y^7*z^2 - 16*x^9 + x*y*z^7 + 2*z^9]) # optional - sage.rings.finite_rings sage: C.quadratic_transform() # optional - sage.rings.finite_rings Scheme morphism: From: Projective Plane Curve over Finite Field of size 17 defined by x^9 + y^7*z^2 + x*y*z^7 + 2*z^9 To: Projective Plane Curve over Finite Field of size 17 defined by 2*x^9*y^7 + x^8*y^6*z^2 + x^9*z^7 + y^7*z^9 Defn: Defined on coordinates by sending (x : y : z) to (y*z : x*z : x*y)
- tangents(P, factor=True)#
Return the tangents of this projective plane curve at the point
P.These are found by homogenizing the tangents of an affine patch of this curve containing
P. The pointPmust 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: P.<x,y,z> = ProjectiveSpace(QQbar, 2) # optional - sage.rings.number_field sage: C = Curve([x^3*y + 2*x^2*y^2 + x*y^3 + x^3*z # optional - sage.rings.number_field ....: + 7*x^2*y*z + 14*x*y^2*z + 9*y^3*z], P) sage: Q = P([0,0,1]) # optional - sage.rings.number_field sage: C.tangents(Q) # optional - sage.rings.number_field [x + 4.147899035704788?*y, x + (1.426050482147607? + 0.3689894074818041?*I)*y, x + (1.426050482147607? - 0.3689894074818041?*I)*y] sage: C.tangents(Q, factor=False) # optional - sage.rings.number_field [6*x^3 + 42*x^2*y + 84*x*y^2 + 54*y^3]
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = P.curve([x^2*y^3*z^4 - y^6*z^3 - 4*x^2*y^4*z^3 - 4*x^4*y^2*z^3 ....: + 3*y^7*z^2 + 10*x^2*y^5*z^2 + 9*x^4*y^3*z^2 + 5*x^6*y*z^2 ....: - 3*y^8*z - 9*x^2*y^6*z - 11*x^4*y^4*z - 7*x^6*y^2*z ....: - 2*x^8*z + y^9 + 2*x^2*y^7 + 3*x^4*y^5 + 4*x^6*y^3 + 2*x^8*y]) sage: Q = P([0,1,1]) sage: C.tangents(Q) [-y + z, 3*x^2 - y^2 + 2*y*z - z^2]
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = P.curve([z^3*x + y^4 - x^2*z^2]) sage: Q = P([1,1,1]) sage: C.tangents(Q) Traceback (most recent call last): ... TypeError: (=(1 : 1 : 1)) is not a point on (=Projective Plane Curve over Rational Field defined by y^4 - x^2*z^2 + x*z^3)
- class sage.schemes.curves.projective_curve.ProjectivePlaneCurve_field(A, f)#
Bases:
ProjectivePlaneCurve,ProjectiveCurve_fieldProjective plane curves over fields.
- arithmetic_genus()#
Return the arithmetic genus of this projective curve.
This is the arithmetic genus \(g_a(C)\) as defined in [Har1977]. For a projective plane curve of degree \(d\), this is simply \((d-1)(d-2)/2\). It need not equal the geometric genus (the genus of the normalization of the curve). This curve must be irreducible.
EXAMPLES:
sage: x,y,z = PolynomialRing(GF(5), 3, 'xyz').gens() # optional - sage.rings.finite_rings sage: C = Curve(y^2*z^7 - x^9 - x*z^8); C # optional - sage.rings.finite_rings Projective Plane Curve over Finite Field of size 5 defined by -x^9 + y^2*z^7 - x*z^8 sage: C.arithmetic_genus() # optional - sage.rings.finite_rings 28 sage: C.genus() # optional - sage.rings.finite_rings 4
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([y^3*x - x^2*y*z - 7*z^4]) sage: C.arithmetic_genus() 3
- 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: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = P.curve(x^2*z - y^3) sage: C.fundamental_group() # optional - sirocco Finitely presented group < x0 | x0^3 >
In the case of number fields, they need to have an embedding into the algebraic field:
sage: a = QQ[x](x^2 + 5).roots(QQbar)[0][0] # optional - sage.rings.number_field sage: a # optional - sage.rings.number_field -2.236067977499790?*I sage: F = NumberField(a.minpoly(), 'a', embedding=a) # optional - sage.rings.number_field sage: P.<x,y,z> = ProjectiveSpace(F, 2) # optional - sage.rings.number_field sage: F.inject_variables() # optional - sage.rings.number_field Defining a sage: C = P.curve(x^2 + a * y^2) # optional - sage.rings.number_field sage: C.fundamental_group() # optional - sirocco # optional - sage.rings.number_field Finitely presented group < x0 | >
Warning
This functionality requires the
siroccopackage to be installed.
- 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 \(\mathbb{P}^{1}\) and this curve, given as a scheme morphism.
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([y^2*z - x^3], P) sage: C.rational_parameterization() Scheme morphism: From: Projective Space of dimension 1 over Rational Field To: Projective Plane Curve over Rational Field defined by -x^3 + y^2*z Defn: Defined on coordinates by sending (s : t) to (s^2*t : s^3 : t^3)
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([x^3 - 4*y*z^2 + x*z^2 - x*y*z], P) sage: C.rational_parameterization() Scheme morphism: From: Projective Space of dimension 1 over Rational Field To: Projective Plane Curve over Rational Field defined by x^3 - x*y*z + x*z^2 - 4*y*z^2 Defn: Defined on coordinates by sending (s : t) to (4*s^2*t + s*t^2 : s^2*t + t^3 : 4*s^3 + s^2*t)
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([x^2 + y^2 + z^2], P) sage: C.rational_parameterization() # optional - sage.rings.number_field Scheme morphism: From: Projective Space of dimension 1 over Number Field in a with defining polynomial a^2 + 1 To: Projective Plane Curve over Number Field in a with defining polynomial a^2 + 1 defined by x^2 + y^2 + z^2 Defn: Defined on coordinates by sending (s : t) to ((-a)*s^2 + (-a)*t^2 : s^2 - t^2 : 2*s*t)
- riemann_surface(**kwargs)#
Return the complex Riemann surface determined by this curve
OUTPUT: A
RiemannSurfaceobject.EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: C = Curve(x^3 + 3*y^3 + 5*z^3) sage: C.riemann_surface() Riemann surface defined by polynomial f = x^3 + 3*y^3 + 5 = 0, with 53 bits of precision
- class sage.schemes.curves.projective_curve.ProjectivePlaneCurve_finite_field(A, f)#
Bases:
ProjectivePlaneCurve_fieldProjective plane curves over finite fields
- rational_points(algorithm='enum', sort=True)#
Return the rational points on this curve.
INPUT:
algorithm– one of'enum'– straightforward enumeration'bn'– via Singular’s brnoeth package.
sort– boolean (default:True); whether the output points should be sorted. If False, the order of the output is non-deterministic.
OUTPUT: A list of all the rational points on the curve, possibly sorted.
Note
The Brill-Noether package does not always work (i.e., the ‘bn’ algorithm. When it fails a RuntimeError exception is raised.
EXAMPLES:
sage: x, y, z = PolynomialRing(GF(5), 3, 'xyz').gens() # optional - sage.rings.finite_rings sage: f = y^2*z^7 - x^9 - x*z^8 # optional - sage.rings.finite_rings sage: C = Curve(f); C # optional - sage.rings.finite_rings Projective Plane Curve over Finite Field of size 5 defined by -x^9 + y^2*z^7 - x*z^8 sage: C.rational_points() # optional - sage.rings.finite_rings [(0 : 0 : 1), (0 : 1 : 0), (2 : 2 : 1), (2 : 3 : 1), (3 : 1 : 1), (3 : 4 : 1)] sage: C = Curve(x - y + z) # optional - sage.rings.finite_rings sage: C.rational_points() # optional - sage.rings.finite_rings [(0 : 1 : 1), (1 : 1 : 0), (1 : 2 : 1), (2 : 3 : 1), (3 : 4 : 1), (4 : 0 : 1)] sage: C = Curve(x*z + z^2) # optional - sage.rings.finite_rings sage: C.rational_points('all') # optional - sage.rings.finite_rings [(0 : 1 : 0), (1 : 0 : 0), (1 : 1 : 0), (2 : 1 : 0), (3 : 1 : 0), (4 : 0 : 1), (4 : 1 : 0), (4 : 1 : 1), (4 : 2 : 1), (4 : 3 : 1), (4 : 4 : 1)]
sage: F = GF(7) # optional - sage.rings.finite_rings sage: P2.<X,Y,Z> = ProjectiveSpace(F, 2) # optional - sage.rings.finite_rings sage: C = Curve(X^3 + Y^3 - Z^3) # optional - sage.rings.finite_rings sage: C.rational_points() # optional - sage.rings.finite_rings [(0 : 1 : 1), (0 : 2 : 1), (0 : 4 : 1), (1 : 0 : 1), (2 : 0 : 1), (3 : 1 : 0), (4 : 0 : 1), (5 : 1 : 0), (6 : 1 : 0)]
sage: F = GF(1237) # optional - sage.rings.finite_rings sage: P2.<X,Y,Z> = ProjectiveSpace(F, 2) # optional - sage.rings.finite_rings sage: C = Curve(X^7 + 7*Y^6*Z + Z^4*X^2*Y*89) # optional - sage.rings.finite_rings sage: len(C.rational_points()) # optional - sage.rings.finite_rings 1237
sage: F = GF(2^6,'a') # optional - sage.rings.finite_rings sage: P2.<X,Y,Z> = ProjectiveSpace(F, 2) # optional - sage.rings.finite_rings sage: C = Curve(X^5 + 11*X*Y*Z^3 + X^2*Y^3 - 13*Y^2*Z^3) # optional - sage.rings.finite_rings sage: len(C.rational_points()) # optional - sage.rings.finite_rings 104
sage: R.<x,y,z> = GF(2)[] # optional - sage.rings.finite_rings sage: f = x^3*y + y^3*z + x*z^3 # optional - sage.rings.finite_rings sage: C = Curve(f); pts = C.rational_points() # optional - sage.rings.finite_rings sage: pts # optional - sage.rings.finite_rings [(0 : 0 : 1), (0 : 1 : 0), (1 : 0 : 0)]
- rational_points_iterator()#
Return a generator object for the rational points on this curve.
INPUT:
self– a projective curve
OUTPUT:
A generator of all the rational points on the curve defined over its base field.
EXAMPLES:
sage: F = GF(37) # optional - sage.rings.finite_rings sage: P2.<X,Y,Z> = ProjectiveSpace(F, 2) # optional - sage.rings.finite_rings sage: C = Curve(X^7 + Y*X*Z^5*55 + Y^7*12) # optional - sage.rings.finite_rings sage: len(list(C.rational_points_iterator())) # optional - sage.rings.finite_rings 37
sage: F = GF(2) # optional - sage.rings.finite_rings sage: P2.<X,Y,Z> = ProjectiveSpace(F, 2) # optional - sage.rings.finite_rings sage: C = Curve(X*Y*Z) # optional - sage.rings.finite_rings sage: a = C.rational_points_iterator() # optional - sage.rings.finite_rings sage: next(a) # optional - sage.rings.finite_rings (1 : 0 : 0) sage: next(a) # optional - sage.rings.finite_rings (0 : 1 : 0) sage: next(a) # optional - sage.rings.finite_rings (1 : 1 : 0) sage: next(a) # optional - sage.rings.finite_rings (0 : 0 : 1) sage: next(a) # optional - sage.rings.finite_rings (1 : 0 : 1) sage: next(a) # optional - sage.rings.finite_rings (0 : 1 : 1) sage: next(a) # optional - sage.rings.finite_rings Traceback (most recent call last): ... StopIteration
sage: F = GF(3^2,'a') # optional - sage.rings.finite_rings sage: P2.<X,Y,Z> = ProjectiveSpace(F, 2) # optional - sage.rings.finite_rings sage: C = Curve(X^3 + 5*Y^2*Z - 33*X*Y*X) # optional - sage.rings.finite_rings sage: b = C.rational_points_iterator() # optional - sage.rings.finite_rings sage: next(b) # optional - sage.rings.finite_rings (0 : 1 : 0) sage: next(b) # optional - sage.rings.finite_rings (0 : 0 : 1) sage: next(b) # optional - sage.rings.finite_rings (2*a + 2 : a : 1) sage: next(b) # optional - sage.rings.finite_rings (2 : a + 1 : 1) sage: next(b) # optional - sage.rings.finite_rings (a + 1 : 2*a + 1 : 1) sage: next(b) # optional - sage.rings.finite_rings (1 : 2 : 1) sage: next(b) # optional - sage.rings.finite_rings (2*a + 2 : 2*a : 1) sage: next(b) # optional - sage.rings.finite_rings (2 : 2*a + 2 : 1) sage: next(b) # optional - sage.rings.finite_rings (a + 1 : a + 2 : 1) sage: next(b) # optional - sage.rings.finite_rings (1 : 1 : 1) sage: next(b) # optional - sage.rings.finite_rings Traceback (most recent call last): ... StopIteration
- riemann_roch_basis(D)#
Return a basis for the Riemann-Roch space corresponding to \(D\).
This uses Singular’s Brill-Noether implementation.
INPUT:
D- a divisor
OUTPUT: A list of function field elements that form a basis of the Riemann-Roch space.
EXAMPLES:
sage: R.<x,y,z> = GF(2)[] # optional - sage.rings.finite_rings sage: f = x^3*y + y^3*z + x*z^3 # optional - sage.rings.finite_rings sage: C = Curve(f); pts = C.rational_points() # optional - sage.rings.finite_rings sage: D = C.divisor([ (4, pts[0]), (4, pts[2]) ]) # optional - sage.rings.finite_rings sage: C.riemann_roch_basis(D) # optional - sage.rings.finite_rings [x/y, 1, z/y, z^2/y^2, z/x, z^2/(x*y)]
sage: R.<x,y,z> = GF(5)[] # optional - sage.rings.finite_rings sage: f = x^7 + y^7 + z^7 # optional - sage.rings.finite_rings sage: C = Curve(f); pts = C.rational_points() # optional - sage.rings.finite_rings sage: D = C.divisor([ (3, pts[0]), (-1,pts[1]), (10, pts[5]) ]) # optional - sage.rings.finite_rings sage: C.riemann_roch_basis(D) # optional - sage.rings.finite_rings [(-2*x + y)/(x + y), (-x + z)/(x + y)]
Note
Currently this only works over prime field and divisors supported on rational points.