def QuarticCurve(F, PP=None, check=False):
"""
Returns the quartic curve defined by the polynomial F.
INPUT:
- F -- a polynomial in three variables, homogeneous of degree 4
- PP -- a projective plane (default:None)
- check -- whether to check for smoothness or not (default:False)
EXAMPLES::
sage: x,y,z=PolynomialRing(QQ,['x','y','z']).gens()
sage: QuarticCurve(x**4+y**4+z**4)
Quartic Curve over Rational Field defined by x^4 + y^4 + z^4
TESTS::
sage: QuarticCurve(x**3+y**3)
Traceback (most recent call last):
...
ValueError: Argument F (=x^3 + y^3) must be a homogeneous polynomial of degree 4
sage: QuarticCurve(x**4+y**4+z**3)
Traceback (most recent call last):
...
ValueError: Argument F (=x^4 + y^4 + z^3) must be a homogeneous polynomial of degree 4
sage: x,y=PolynomialRing(QQ,['x','y']).gens()
sage: QuarticCurve(x**4+y**4)
Traceback (most recent call last):
...
ValueError: Argument F (=x^4 + y^4) must be a polynomial in 3 variables
"""
if not is_MPolynomial(F):
raise ValueError("Argument F (=%s) must be a multivariate polynomial"%F)
P = F.parent()
if not P.ngens() == 3:
raise ValueError("Argument F (=%s) must be a polynomial in 3 variables"%F)
if not(F.is_homogeneous() and F.degree()==4):
raise ValueError("Argument F (=%s) must be a homogeneous polynomial of degree 4"%F)
if not PP is None:
if not is_ProjectiveSpace(PP) and PP.dimension == 2:
raise ValueError("Argument PP (=%s) must be a projective plane"%PP)
else:
PP = ProjectiveSpace(P)
if check:
raise NotImplementedError("Argument checking (for nonsingularity) is not implemented.")
return QuarticCurve_generic(PP, F)
def Curve(F):
"""
Return the plane or space curve defined by `F`, where
`F` can be either a multivariate polynomial, a list or
tuple of polynomials, or an algebraic scheme.
If `F` is in two variables the curve is affine, and if it
is homogenous in `3` variables, then the curve is
projective.
EXAMPLE: A projective plane curve
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3); C
Projective Curve over Rational Field defined by x^3 + y^3 + z^3
sage: C.genus()
1
EXAMPLE: Affine plane curves
::
sage: x,y = GF(7)['x,y'].gens()
sage: C = Curve(y^2 + x^3 + x^10); C
Affine Curve over Finite Field of size 7 defined by x^10 + x^3 + y^2
sage: C.genus()
0
sage: x, y = QQ['x,y'].gens()
sage: Curve(x^3 + y^3 + 1)
Affine Curve over Rational Field defined by x^3 + y^3 + 1
EXAMPLE: A projective space curve
::
sage: x,y,z,w = QQ['x,y,z,w'].gens()
sage: C = Curve([x^3 + y^3 - z^3 - w^3, x^5 - y*z^4]); C
Projective Space Curve over Rational Field defined by x^3 + y^3 - z^3 - w^3, x^5 - y*z^4
sage: C.genus()
13
EXAMPLE: An affine space curve
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve([y^2 + x^3 + x^10 + z^7, x^2 + y^2]); C
Affine Space Curve over Rational Field defined by x^10 + z^7 + x^3 + y^2, x^2 + y^2
sage: C.genus()
47
EXAMPLE: We can also make non-reduced non-irreducible curves.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve((x-y)*(x+y))
Projective Conic Curve over Rational Field defined by x^2 - y^2
sage: Curve((x-y)^2*(x+y)^2)
Projective Curve over Rational Field defined by x^4 - 2*x^2*y^2 + y^4
EXAMPLE: A union of curves is a curve.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3)
sage: D = Curve(x^4 + y^4 + z^4)
sage: C.union(D)
Projective Curve over Rational Field defined by
x^7 + x^4*y^3 + x^3*y^4 + y^7 + x^4*z^3 + y^4*z^3 + x^3*z^4 + y^3*z^4 + z^7
The intersection is not a curve, though it is a scheme.
::
sage: X = C.intersection(D); X
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x^3 + y^3 + z^3,
x^4 + y^4 + z^4
Note that the intersection has dimension `0`.
::
sage: X.dimension()
0
sage: I = X.defining_ideal(); I
Ideal (x^3 + y^3 + z^3, x^4 + y^4 + z^4) of Multivariate Polynomial Ring in x, y, z over Rational Field
EXAMPLE: In three variables, the defining equation must be
homogeneous.
If the parent polynomial ring is in three variables, then the
defining ideal must be homogeneous.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve(x^2+y^2)
Projective Conic Curve over Rational Field defined by x^2 + y^2
sage: Curve(x^2+y^2+z)
Traceback (most recent call last):
...
TypeError: x^2 + y^2 + z is not a homogeneous polynomial!
The defining polynomial must always be nonzero::
sage: P1.<x,y> = ProjectiveSpace(1,GF(5))
sage: Curve(0*x)
Traceback (most recent call last):
...
ValueError: defining polynomial of curve must be nonzero
"""
if is_AlgebraicScheme(F):
return Curve(F.defining_polynomials())
if isinstance(F, (list, tuple)):
if len(F) == 1:
return Curve(F[0])
F = Sequence(F)
P = F.universe()
if not is_MPolynomialRing(P):
raise TypeError("universe of F must be a multivariate polynomial ring")
for f in F:
if not f.is_homogeneous():
A = AffineSpace(P.ngens(), P.base_ring())
A._coordinate_ring = P
return AffineSpaceCurve_generic(A, F)
A = ProjectiveSpace(P.ngens()-1, P.base_ring())
A._coordinate_ring = P
return ProjectiveSpaceCurve_generic(A, F)
if not is_MPolynomial(F):
raise TypeError("F (=%s) must be a multivariate polynomial"%F)
P = F.parent()
k = F.base_ring()
if F.parent().ngens() == 2:
if F == 0:
raise ValueError("defining polynomial of curve must be nonzero")
A2 = AffineSpace(2, P.base_ring())
A2._coordinate_ring = P
if is_FiniteField(k):
if k.is_prime_field():
return AffineCurve_prime_finite_field(A2, F)
else:
return AffineCurve_finite_field(A2, F)
else:
return AffineCurve_generic(A2, F)
elif F.parent().ngens() == 3:
if F == 0:
raise ValueError("defining polynomial of curve must be nonzero")
P2 = ProjectiveSpace(2, P.base_ring())
P2._coordinate_ring = P
if F.total_degree() == 2 and k.is_field():
return Conic(F)
if is_FiniteField(k):
if k.is_prime_field():
return ProjectiveCurve_prime_finite_field(P2, F)
else:
return ProjectiveCurve_finite_field(P2, F)
else:
return ProjectiveCurve_generic(P2, F)
else:
raise TypeError("Number of variables of F (=%s) must be 2 or 3"%F)
def Curve(F, A=None):
"""
Return the plane or space curve defined by ``F``, where
``F`` can be either a multivariate polynomial, a list or
tuple of polynomials, or an algebraic scheme.
If no ambient space is passed in for ``A``, and if ``F`` is not
an algebraic scheme, a new ambient space is constructed.
Also not specifying an ambient space will cause the curve to be defined
in either affine or projective space based on properties of ``F``. In
particular, if ``F`` contains a nonhomogenous polynomial, the curve is
affine, and if ``F`` consists of homogenous polynomials, then the curve
is projective.
INPUT:
- ``F`` -- a multivariate polynomial, or a list or tuple of polynomials, or an algebraic scheme.
- ``A`` -- (default: None) an ambient space in which to create the curve.
EXAMPLES: A projective plane curve
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3); C
Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3
sage: C.genus()
1
EXAMPLES: Affine plane curves
::
sage: x,y = GF(7)['x,y'].gens()
sage: C = Curve(y^2 + x^3 + x^10); C
Affine Plane Curve over Finite Field of size 7 defined by x^10 + x^3 + y^2
sage: C.genus()
0
sage: x, y = QQ['x,y'].gens()
sage: Curve(x^3 + y^3 + 1)
Affine Plane Curve over Rational Field defined by x^3 + y^3 + 1
EXAMPLES: A projective space curve
::
sage: x,y,z,w = QQ['x,y,z,w'].gens()
sage: C = Curve([x^3 + y^3 - z^3 - w^3, x^5 - y*z^4]); C
Projective Curve over Rational Field defined by x^3 + y^3 - z^3 - w^3, x^5 - y*z^4
sage: C.genus()
13
EXAMPLES: An affine space curve
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve([y^2 + x^3 + x^10 + z^7, x^2 + y^2]); C
Affine Curve over Rational Field defined by x^10 + z^7 + x^3 + y^2, x^2 + y^2
sage: C.genus()
47
EXAMPLES: We can also make non-reduced non-irreducible curves.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve((x-y)*(x+y))
Projective Conic Curve over Rational Field defined by x^2 - y^2
sage: Curve((x-y)^2*(x+y)^2)
Projective Plane Curve over Rational Field defined by x^4 - 2*x^2*y^2 + y^4
EXAMPLES: A union of curves is a curve.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3)
sage: D = Curve(x^4 + y^4 + z^4)
sage: C.union(D)
Projective Plane Curve over Rational Field defined by
x^7 + x^4*y^3 + x^3*y^4 + y^7 + x^4*z^3 + y^4*z^3 + x^3*z^4 + y^3*z^4 + z^7
The intersection is not a curve, though it is a scheme.
::
sage: X = C.intersection(D); X
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x^3 + y^3 + z^3,
x^4 + y^4 + z^4
Note that the intersection has dimension `0`.
::
sage: X.dimension()
0
sage: I = X.defining_ideal(); I
Ideal (x^3 + y^3 + z^3, x^4 + y^4 + z^4) of Multivariate Polynomial Ring in x, y, z over Rational Field
EXAMPLES: In three variables, the defining equation must be
homogeneous.
If the parent polynomial ring is in three variables, then the
defining ideal must be homogeneous.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve(x^2+y^2)
Projective Conic Curve over Rational Field defined by x^2 + y^2
sage: Curve(x^2+y^2+z)
Traceback (most recent call last):
...
TypeError: x^2 + y^2 + z is not a homogeneous polynomial
The defining polynomial must always be nonzero::
sage: P1.<x,y> = ProjectiveSpace(1,GF(5))
sage: Curve(0*x)
Traceback (most recent call last):
...
ValueError: defining polynomial of curve must be nonzero
::
sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: C = Curve([y - x^2, z - x^3], A)
sage: A == C.ambient_space()
True
"""
if not A is None:
if not isinstance(F, (list, tuple)):
return Curve([F], A)
if not is_AmbientSpace(A):
raise TypeError("A (=%s) must be either an affine or projective space"%A)
if not all([f.parent() == A.coordinate_ring() for f in F]):
raise TypeError("F (=%s) must be a list or tuple of polynomials of the coordinate ring of " \
"A (=%s)"%(F, A))
n = A.dimension_relative()
if n < 2:
raise TypeError("A (=%s) must be either an affine or projective space of dimension > 1"%A)
# there is no dimension check when initializing a plane curve, so check here that F consists
# of a single nonconstant polynomial
if n == 2:
if len(F) != 1 or F[0] == 0 or not is_MPolynomial(F[0]):
raise TypeError("F (=%s) must consist of a single nonconstant polynomial to define a plane curve"%(F,))
if is_AffineSpace(A):
if n > 2:
return AffineCurve(A, F)
k = A.base_ring()
if is_FiniteField(k):
if k.is_prime_field():
return AffinePlaneCurve_prime_finite_field(A, F[0])
return AffinePlaneCurve_finite_field(A, F[0])
return AffinePlaneCurve(A, F[0])
elif is_ProjectiveSpace(A):
if not all([f.is_homogeneous() for f in F]):
raise TypeError("polynomials defining a curve in a projective space must be homogeneous")
if n > 2:
return ProjectiveCurve(A, F)
k = A.base_ring()
if is_FiniteField(k):
if k.is_prime_field():
return ProjectivePlaneCurve_prime_finite_field(A, F[0])
return ProjectivePlaneCurve_finite_field(A, F[0])
return ProjectivePlaneCurve(A, F[0])
if is_AlgebraicScheme(F):
return Curve(F.defining_polynomials(), F.ambient_space())
if isinstance(F, (list, tuple)):
if len(F) == 1:
return Curve(F[0])
F = Sequence(F)
P = F.universe()
if not is_MPolynomialRing(P):
raise TypeError("universe of F must be a multivariate polynomial ring")
for f in F:
if not f.is_homogeneous():
A = AffineSpace(P.ngens(), P.base_ring())
A._coordinate_ring = P
return AffineCurve(A, F)
A = ProjectiveSpace(P.ngens()-1, P.base_ring())
A._coordinate_ring = P
return ProjectiveCurve(A, F)
if not is_MPolynomial(F):
raise TypeError("F (=%s) must be a multivariate polynomial"%F)
P = F.parent()
k = F.base_ring()
if F.parent().ngens() == 2:
if F == 0:
raise ValueError("defining polynomial of curve must be nonzero")
A2 = AffineSpace(2, P.base_ring())
A2._coordinate_ring = P
if is_FiniteField(k):
if k.is_prime_field():
return AffinePlaneCurve_prime_finite_field(A2, F)
else:
return AffinePlaneCurve_finite_field(A2, F)
else:
return AffinePlaneCurve(A2, F)
elif F.parent().ngens() == 3:
if F == 0:
raise ValueError("defining polynomial of curve must be nonzero")
P2 = ProjectiveSpace(2, P.base_ring())
P2._coordinate_ring = P
if F.total_degree() == 2 and k.is_field():
return Conic(F)
if is_FiniteField(k):
if k.is_prime_field():
return ProjectivePlaneCurve_prime_finite_field(P2, F)
else:
return ProjectivePlaneCurve_finite_field(P2, F)
else:
return ProjectivePlaneCurve(P2, F)
else:
raise TypeError("Number of variables of F (=%s) must be 2 or 3"%F)
def Curve(F, A=None):
"""
Return the plane or space curve defined by ``F``, where ``F`` can be either
a multivariate polynomial, a list or tuple of polynomials, or an algebraic
scheme.
If no ambient space is passed in for ``A``, and if ``F`` is not an
algebraic scheme, a new ambient space is constructed.
Also not specifying an ambient space will cause the curve to be defined in
either affine or projective space based on properties of ``F``. In
particular, if ``F`` contains a nonhomogenous polynomial, the curve is
affine, and if ``F`` consists of homogenous polynomials, then the curve is
projective.
INPUT:
- ``F`` -- a multivariate polynomial, or a list or tuple of polynomials, or an algebraic scheme.
- ``A`` -- (default: None) an ambient space in which to create the curve.
EXAMPLES: A projective plane curve. ::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3); C
Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3
sage: C.genus()
1
Affine plane curves. ::
sage: x,y = GF(7)['x,y'].gens()
sage: C = Curve(y^2 + x^3 + x^10); C
Affine Plane Curve over Finite Field of size 7 defined by x^10 + x^3 + y^2
sage: C.genus()
0
sage: x, y = QQ['x,y'].gens()
sage: Curve(x^3 + y^3 + 1)
Affine Plane Curve over Rational Field defined by x^3 + y^3 + 1
A projective space curve. ::
sage: x,y,z,w = QQ['x,y,z,w'].gens()
sage: C = Curve([x^3 + y^3 - z^3 - w^3, x^5 - y*z^4]); C
Projective Curve over Rational Field defined by x^3 + y^3 - z^3 - w^3, x^5 - y*z^4
sage: C.genus()
13
An affine space curve. ::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve([y^2 + x^3 + x^10 + z^7, x^2 + y^2]); C
Affine Curve over Rational Field defined by x^10 + z^7 + x^3 + y^2, x^2 + y^2
sage: C.genus()
47
We can also make non-reduced non-irreducible curves. ::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve((x-y)*(x+y))
Projective Conic Curve over Rational Field defined by x^2 - y^2
sage: Curve((x-y)^2*(x+y)^2)
Projective Plane Curve over Rational Field defined by x^4 - 2*x^2*y^2 + y^4
A union of curves is a curve. ::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3)
sage: D = Curve(x^4 + y^4 + z^4)
sage: C.union(D)
Projective Plane Curve over Rational Field defined by
x^7 + x^4*y^3 + x^3*y^4 + y^7 + x^4*z^3 + y^4*z^3 + x^3*z^4 + y^3*z^4 + z^7
The intersection is not a curve, though it is a scheme. ::
sage: X = C.intersection(D); X
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x^3 + y^3 + z^3,
x^4 + y^4 + z^4
Note that the intersection has dimension 0. ::
sage: X.dimension()
0
sage: I = X.defining_ideal(); I
Ideal (x^3 + y^3 + z^3, x^4 + y^4 + z^4) of Multivariate Polynomial Ring in x, y, z over Rational Field
If only a polynomial in three variables is given, then it must be
homogeneous such that a projective curve is constructed. ::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve(x^2+y^2)
Projective Conic Curve over Rational Field defined by x^2 + y^2
sage: Curve(x^2+y^2+z)
Traceback (most recent call last):
...
TypeError: x^2 + y^2 + z is not a homogeneous polynomial
An ambient space can be specified to construct a space curve in an affine
or a projective space. ::
sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: C = Curve([y - x^2, z - x^3], A)
sage: C
Affine Curve over Rational Field defined by -x^2 + y, -x^3 + z
sage: A == C.ambient_space()
True
The defining polynomial must be nonzero unless the ambient space itself is
of dimension 1. ::
sage: P1.<x,y> = ProjectiveSpace(1,GF(5))
sage: S = P1.coordinate_ring()
sage: Curve(S(0), P1)
Projective Line over Finite Field of size 5
sage: Curve(P1)
Projective Line over Finite Field of size 5
::
sage: A1.<x> = AffineSpace(1, QQ)
sage: R = A1.coordinate_ring()
sage: Curve(R(0), A1)
Affine Line over Rational Field
sage: Curve(A1)
Affine Line over Rational Field
"""
if A is None:
if is_AmbientSpace(F) and F.dimension() == 1:
return Curve(F.coordinate_ring().zero(), F)
if is_AlgebraicScheme(F):
return Curve(F.defining_polynomials(), F.ambient_space())
if isinstance(F, (list, tuple)):
P = Sequence(F).universe()
if not is_MPolynomialRing(P):
raise TypeError("universe of F must be a multivariate polynomial ring")
for f in F:
if not f.is_homogeneous():
A = AffineSpace(P.ngens(), P.base_ring(), names=P.variable_names())
A._coordinate_ring = P
break
else:
A = ProjectiveSpace(P.ngens()-1, P.base_ring(), names=P.variable_names())
A._coordinate_ring = P
elif is_MPolynomial(F): # define a plane curve
P = F.parent()
k = F.base_ring()
if not k.is_field():
if k.is_integral_domain(): # upgrade to a field
P = P.change_ring(k.fraction_field())
F = P(F)
k = F.base_ring()
else:
raise TypeError("not a multivariate polynomial over a field or an integral domain")
if F.parent().ngens() == 2:
if F == 0:
raise ValueError("defining polynomial of curve must be nonzero")
A = AffineSpace(2, P.base_ring(), names=P.variable_names())
A._coordinate_ring = P
elif F.parent().ngens() == 3:
if F == 0:
raise ValueError("defining polynomial of curve must be nonzero")
# special case: construct a conic curve
if F.total_degree() == 2 and k.is_field():
return Conic(k, F)
A = ProjectiveSpace(2, P.base_ring(), names=P.variable_names())
A._coordinate_ring = P
elif F.parent().ngens() == 1:
if not F.is_zero():
raise ValueError("defining polynomial of curve must be zero "
"if the ambient space is of dimension 1")
A = AffineSpace(1, P.base_ring(), names=P.variable_names())
A._coordinate_ring = P
else:
raise TypeError("number of variables of F (={}) must be 2 or 3".format(F))
F = [F]
else:
raise TypeError("F (={}) must be a multivariate polynomial".format(F))
else:
if not is_AmbientSpace(A):
raise TypeError("ambient space must be either an affine or projective space")
if not isinstance(F, (list, tuple)):
F = [F]
if not all(f.parent() == A.coordinate_ring() for f in F):
raise TypeError("need a list of polynomials of the coordinate ring of {}".format(A))
n = A.dimension_relative()
if n < 1:
raise TypeError("ambient space should be an affine or projective space of positive dimension")
k = A.base_ring()
if is_AffineSpace(A):
if n != 2:
if is_FiniteField(k):
if A.coordinate_ring().ideal(F).is_prime():
return IntegralAffineCurve_finite_field(A, F)
if k in Fields():
if k == QQ and A.coordinate_ring().ideal(F).is_prime():
return IntegralAffineCurve(A, F)
return AffineCurve_field(A, F)
return AffineCurve(A, F)
if not (len(F) == 1 and F[0] != 0 and F[0].degree() > 0):
raise TypeError("need a single nonconstant polynomial to define a plane curve")
F = F[0]
if is_FiniteField(k):
if _is_irreducible_and_reduced(F):
return IntegralAffinePlaneCurve_finite_field(A, F)
return AffinePlaneCurve_finite_field(A, F)
if k in Fields():
if k == QQ and _is_irreducible_and_reduced(F):
return IntegralAffinePlaneCurve(A, F)
return AffinePlaneCurve_field(A, F)
return AffinePlaneCurve(A, F)
elif is_ProjectiveSpace(A):
if n != 2:
if not all(f.is_homogeneous() for f in F):
raise TypeError("polynomials defining a curve in a projective space must be homogeneous")
if is_FiniteField(k):
if A.coordinate_ring().ideal(F).is_prime():
return IntegralProjectiveCurve_finite_field(A, F)
if k in Fields():
if k == QQ and A.coordinate_ring().ideal(F).is_prime():
return IntegralProjectiveCurve(A, F)
return ProjectiveCurve_field(A, F)
return ProjectiveCurve(A, F)
# There is no dimension check when initializing a plane curve, so check
# here that F consists of a single nonconstant polynomial.
if not (len(F) == 1 and F[0] != 0 and F[0].degree() > 0):
raise TypeError("need a single nonconstant polynomial to define a plane curve")
F = F[0]
if not F.is_homogeneous():
raise TypeError("{} is not a homogeneous polynomial".format(F))
if is_FiniteField(k):
if _is_irreducible_and_reduced(F):
return IntegralProjectivePlaneCurve_finite_field(A, F)
return ProjectivePlaneCurve_finite_field(A, F)
if k in Fields():
if k == QQ and _is_irreducible_and_reduced(F):
return IntegralProjectivePlaneCurve(A, F)
return ProjectivePlaneCurve_field(A, F)
return ProjectivePlaneCurve(A, F)
else:
raise TypeError('ambient space neither affine nor projective')
def Curve(F, A=None):
"""
Return the plane or space curve defined by ``F``, where
``F`` can be either a multivariate polynomial, a list or
tuple of polynomials, or an algebraic scheme.
If no ambient space is passed in for ``A``, and if ``F`` is not
an algebraic scheme, a new ambient space is constructed.
Also not specifying an ambient space will cause the curve to be defined
in either affine or projective space based on properties of ``F``. In
particular, if ``F`` contains a nonhomogenous polynomial, the curve is
affine, and if ``F`` consists of homogenous polynomials, then the curve
is projective.
INPUT:
- ``F`` -- a multivariate polynomial, or a list or tuple of polynomials, or an algebraic scheme.
- ``A`` -- (default: None) an ambient space in which to create the curve.
EXAMPLE: A projective plane curve
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3); C
Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3
sage: C.genus()
1
EXAMPLE: Affine plane curves
::
sage: x,y = GF(7)['x,y'].gens()
sage: C = Curve(y^2 + x^3 + x^10); C
Affine Plane Curve over Finite Field of size 7 defined by x^10 + x^3 + y^2
sage: C.genus()
0
sage: x, y = QQ['x,y'].gens()
sage: Curve(x^3 + y^3 + 1)
Affine Plane Curve over Rational Field defined by x^3 + y^3 + 1
EXAMPLE: A projective space curve
::
sage: x,y,z,w = QQ['x,y,z,w'].gens()
sage: C = Curve([x^3 + y^3 - z^3 - w^3, x^5 - y*z^4]); C
Projective Curve over Rational Field defined by x^3 + y^3 - z^3 - w^3, x^5 - y*z^4
sage: C.genus()
13
EXAMPLE: An affine space curve
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve([y^2 + x^3 + x^10 + z^7, x^2 + y^2]); C
Affine Curve over Rational Field defined by x^10 + z^7 + x^3 + y^2, x^2 + y^2
sage: C.genus()
47
EXAMPLE: We can also make non-reduced non-irreducible curves.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve((x-y)*(x+y))
Projective Conic Curve over Rational Field defined by x^2 - y^2
sage: Curve((x-y)^2*(x+y)^2)
Projective Plane Curve over Rational Field defined by x^4 - 2*x^2*y^2 + y^4
EXAMPLE: A union of curves is a curve.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3)
sage: D = Curve(x^4 + y^4 + z^4)
sage: C.union(D)
Projective Plane Curve over Rational Field defined by
x^7 + x^4*y^3 + x^3*y^4 + y^7 + x^4*z^3 + y^4*z^3 + x^3*z^4 + y^3*z^4 + z^7
The intersection is not a curve, though it is a scheme.
::
sage: X = C.intersection(D); X
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x^3 + y^3 + z^3,
x^4 + y^4 + z^4
Note that the intersection has dimension `0`.
::
sage: X.dimension()
0
sage: I = X.defining_ideal(); I
Ideal (x^3 + y^3 + z^3, x^4 + y^4 + z^4) of Multivariate Polynomial Ring in x, y, z over Rational Field
EXAMPLE: In three variables, the defining equation must be
homogeneous.
If the parent polynomial ring is in three variables, then the
defining ideal must be homogeneous.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve(x^2+y^2)
Projective Conic Curve over Rational Field defined by x^2 + y^2
sage: Curve(x^2+y^2+z)
Traceback (most recent call last):
...
TypeError: x^2 + y^2 + z is not a homogeneous polynomial
The defining polynomial must always be nonzero::
sage: P1.<x,y> = ProjectiveSpace(1,GF(5))
sage: Curve(0*x)
Traceback (most recent call last):
...
ValueError: defining polynomial of curve must be nonzero
::
sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: C = Curve([y - x^2, z - x^3], A)
sage: A == C.ambient_space()
True
"""
if not A is None:
if not isinstance(F, (list, tuple)):
return Curve([F], A)
if not is_AmbientSpace(A):
raise TypeError(
"A (=%s) must be either an affine or projective space" % A)
if not all([f.parent() == A.coordinate_ring() for f in F]):
raise TypeError("F (=%s) must be a list or tuple of polynomials of the coordinate ring of " \
"A (=%s)"%(F, A))
n = A.dimension_relative()
if n < 2:
raise TypeError(
"A (=%s) must be either an affine or projective space of dimension > 1"
% A)
# there is no dimension check when initializing a plane curve, so check here that F consists
# of a single nonconstant polynomial
if n == 2:
if len(F) != 1 or F[0] == 0 or not is_MPolynomial(F[0]):
raise TypeError(
"F (=%s) must consist of a single nonconstant polynomial to define a plane curve"
% (F, ))
if is_AffineSpace(A):
if n > 2:
return AffineCurve(A, F)
k = A.base_ring()
if is_FiniteField(k):
if k.is_prime_field():
return AffinePlaneCurve_prime_finite_field(A, F[0])
return AffinePlaneCurve_finite_field(A, F[0])
return AffinePlaneCurve(A, F[0])
elif is_ProjectiveSpace(A):
if not all([f.is_homogeneous() for f in F]):
raise TypeError(
"polynomials defining a curve in a projective space must be homogeneous"
)
if n > 2:
return ProjectiveCurve(A, F)
k = A.base_ring()
if is_FiniteField(k):
if k.is_prime_field():
return ProjectivePlaneCurve_prime_finite_field(A, F[0])
return ProjectivePlaneCurve_finite_field(A, F[0])
return ProjectivePlaneCurve(A, F[0])
if is_AlgebraicScheme(F):
return Curve(F.defining_polynomials(), F.ambient_space())
if isinstance(F, (list, tuple)):
if len(F) == 1:
return Curve(F[0])
F = Sequence(F)
P = F.universe()
if not is_MPolynomialRing(P):
raise TypeError(
"universe of F must be a multivariate polynomial ring")
for f in F:
if not f.is_homogeneous():
A = AffineSpace(P.ngens(), P.base_ring())
A._coordinate_ring = P
return AffineCurve(A, F)
A = ProjectiveSpace(P.ngens() - 1, P.base_ring())
A._coordinate_ring = P
return ProjectiveCurve(A, F)
if not is_MPolynomial(F):
raise TypeError("F (=%s) must be a multivariate polynomial" % F)
P = F.parent()
k = F.base_ring()
if F.parent().ngens() == 2:
if F == 0:
raise ValueError("defining polynomial of curve must be nonzero")
A2 = AffineSpace(2, P.base_ring())
A2._coordinate_ring = P
if is_FiniteField(k):
if k.is_prime_field():
return AffinePlaneCurve_prime_finite_field(A2, F)
else:
return AffinePlaneCurve_finite_field(A2, F)
else:
return AffinePlaneCurve(A2, F)
elif F.parent().ngens() == 3:
if F == 0:
raise ValueError("defining polynomial of curve must be nonzero")
P2 = ProjectiveSpace(2, P.base_ring())
P2._coordinate_ring = P
if F.total_degree() == 2 and k.is_field():
return Conic(F)
if is_FiniteField(k):
if k.is_prime_field():
return ProjectivePlaneCurve_prime_finite_field(P2, F)
else:
return ProjectivePlaneCurve_finite_field(P2, F)
else:
return ProjectivePlaneCurve(P2, F)
else:
raise TypeError("Number of variables of F (=%s) must be 2 or 3" % F)
def Curve(F):
"""
Return the plane or space curve defined by `F`, where
`F` can be either a multivariate polynomial, a list or
tuple of polynomials, or an algebraic scheme.
If `F` is in two variables the curve is affine, and if it
is homogenous in `3` variables, then the curve is
projective.
EXAMPLE: A projective plane curve
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3); C
Projective Curve over Rational Field defined by x^3 + y^3 + z^3
sage: C.genus()
1
EXAMPLE: Affine plane curves
::
sage: x,y = GF(7)['x,y'].gens()
sage: C = Curve(y^2 + x^3 + x^10); C
Affine Curve over Finite Field of size 7 defined by x^10 + x^3 + y^2
sage: C.genus()
0
sage: x, y = QQ['x,y'].gens()
sage: Curve(x^3 + y^3 + 1)
Affine Curve over Rational Field defined by x^3 + y^3 + 1
EXAMPLE: A projective space curve
::
sage: x,y,z,w = QQ['x,y,z,w'].gens()
sage: C = Curve([x^3 + y^3 - z^3 - w^3, x^5 - y*z^4]); C
Projective Space Curve over Rational Field defined by x^3 + y^3 - z^3 - w^3, x^5 - y*z^4
sage: C.genus()
13
EXAMPLE: An affine space curve
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve([y^2 + x^3 + x^10 + z^7, x^2 + y^2]); C
Affine Space Curve over Rational Field defined by x^10 + z^7 + x^3 + y^2, x^2 + y^2
sage: C.genus()
47
EXAMPLE: We can also make non-reduced non-irreducible curves.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve((x-y)*(x+y))
Projective Conic Curve over Rational Field defined by x^2 - y^2
sage: Curve((x-y)^2*(x+y)^2)
Projective Curve over Rational Field defined by x^4 - 2*x^2*y^2 + y^4
EXAMPLE: A union of curves is a curve.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3)
sage: D = Curve(x^4 + y^4 + z^4)
sage: C.union(D)
Projective Curve over Rational Field defined by
x^7 + x^4*y^3 + x^3*y^4 + y^7 + x^4*z^3 + y^4*z^3 + x^3*z^4 + y^3*z^4 + z^7
The intersection is not a curve, though it is a scheme.
::
sage: X = C.intersection(D); X
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x^3 + y^3 + z^3,
x^4 + y^4 + z^4
Note that the intersection has dimension `0`.
::
sage: X.dimension()
0
sage: I = X.defining_ideal(); I
Ideal (x^3 + y^3 + z^3, x^4 + y^4 + z^4) of Multivariate Polynomial Ring in x, y, z over Rational Field
EXAMPLE: In three variables, the defining equation must be
homogeneous.
If the parent polynomial ring is in three variables, then the
defining ideal must be homogeneous.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve(x^2+y^2)
Projective Conic Curve over Rational Field defined by x^2 + y^2
sage: Curve(x^2+y^2+z)
Traceback (most recent call last):
...
TypeError: x^2 + y^2 + z is not a homogeneous polynomial!
The defining polynomial must always be nonzero::
sage: P1.<x,y> = ProjectiveSpace(1,GF(5))
sage: Curve(0*x)
Traceback (most recent call last):
...
ValueError: defining polynomial of curve must be nonzero
"""
if is_AlgebraicScheme(F):
return Curve(F.defining_polynomials())
if isinstance(F, (list, tuple)):
if len(F) == 1:
return Curve(F[0])
F = Sequence(F)
P = F.universe()
if not is_MPolynomialRing(P):
raise TypeError, "universe of F must be a multivariate polynomial ring"
for f in F:
if not f.is_homogeneous():
A = AffineSpace(P.ngens(), P.base_ring())
A._coordinate_ring = P
return AffineSpaceCurve_generic(A, F)
A = ProjectiveSpace(P.ngens() - 1, P.base_ring())
A._coordinate_ring = P
return ProjectiveSpaceCurve_generic(A, F)
if not is_MPolynomial(F):
raise TypeError, "F (=%s) must be a multivariate polynomial" % F
P = F.parent()
k = F.base_ring()
if F.parent().ngens() == 2:
if F == 0:
raise ValueError, "defining polynomial of curve must be nonzero"
A2 = AffineSpace(2, P.base_ring())
A2._coordinate_ring = P
if is_FiniteField(k):
if k.is_prime_field():
return AffineCurve_prime_finite_field(A2, F)
else:
return AffineCurve_finite_field(A2, F)
else:
return AffineCurve_generic(A2, F)
elif F.parent().ngens() == 3:
if F == 0:
raise ValueError, "defining polynomial of curve must be nonzero"
P2 = ProjectiveSpace(2, P.base_ring())
P2._coordinate_ring = P
if F.total_degree() == 2 and k.is_field():
return Conic(F)
if is_FiniteField(k):
if k.is_prime_field():
return ProjectiveCurve_prime_finite_field(P2, F)
else:
return ProjectiveCurve_finite_field(P2, F)
else:
return ProjectiveCurve_generic(P2, F)
else:
raise TypeError, "Number of variables of F (=%s) must be 2 or 3" % F