from sage.misc.all import latex
from sage.schemes.generic.algebraic_scheme import (
AlgebraicScheme_subscheme, AlgebraicScheme_subscheme_projective)
from sage.schemes.generic.divisor_group import DivisorGroup
from sage.schemes.generic.divisor import Divisor_curve
class Curve_generic(AlgebraicScheme_subscheme):
r"""
EXAMPLES::
sage: A.<x,y,z> = AffineSpace(QQ,3)
sage: C = Curve([x-y,z-2])
sage: loads(C.dumps()) == C
True
"""
def _repr_(self):
"""
EXAMPLES::
sage: A.<x,y,z> = AffineSpace(QQ,3)
sage: C = Curve([x-y,z-2])
sage: C
Affine Space Curve over Rational Field defined by x - y, z - 2
sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: C = Curve(x-y)
sage: C
Projective Curve over Rational Field defined by x - y
"""
return "%s Curve over %s defined by %s"%(
self._repr_type(), self.base_ring(), ', '.join([str(x) for x in self.defining_polynomials()]))
def _repr_type(self):
return "Generic"
def _latex_(self):
"""
EXAMPLES:
sage: x,y,z = PolynomialRing(QQ, 3, names='x,y,z').gens()
sage: C = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2)
sage: latex(C)
- x^{3} + y^{2} z - 17 x z^{2} + y z^{2}
sage: A2 = AffineSpace(2, QQ, names=['x','y'])
sage: x, y = A2.coordinate_ring().gens()
sage: C = Curve(y^2 - x^3 - 17*x + y)
sage: latex(C)
- x^{3} + y^{2} - 17 x + y
"""
return latex(self.defining_polynomial())
def defining_polynomial(self):
return self.defining_polynomials()[0]
def divisor_group(self, base_ring=None):
r"""
Return the divisor group of the curve.
INPUT:
- ``base_ring`` -- the base ring of the divisor
group. Usually, this is `\ZZ` (default) or `\QQ`.
OUTPUT:
The divisor group of the curve.
EXAMPLES::
sage: x,y,z = PolynomialRing(QQ, 3, names='x,y,z').gens()
sage: C = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2)
sage: Cp = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2)
sage: C.divisor_group() is Cp.divisor_group()
True
"""
return DivisorGroup(self, base_ring)
def divisor(self, v, base_ring=None, check=True, reduce=True):
r"""
Return the divisor specified by ``v``.
.. WARNING::
The coefficients of the divisor must be in the base ring
and the terms must be reduced. If you set ``check=False``
and/or ``reduce=False`` it is your responsibility to pass
a valid object ``v``.
"""
return Divisor_curve(v, check=check, reduce=reduce, parent=self.divisor_group(base_ring))
def genus(self):
"""
The geometric genus of the curve.
"""
return self.geometric_genus()
def geometric_genus(self):
r"""
The geometric genus of the curve, which is by definition the
genus of the normalization of the projective closure of the
curve over the algebraic closure of the base field; the base
field must be a prime field.
\note{Calls Singular's genus command.}
EXAMPLE:
Examples of projective curves.
sage: P2 = ProjectiveSpace(2, GF(5), names=['x','y','z'])
sage: x, y, z = P2.coordinate_ring().gens()
sage: C = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2)
sage: C.geometric_genus()
1
sage: C = Curve(y^2*z - x^3)
sage: C.geometric_genus()
0
sage: C = Curve(x^10 + y^7*z^3 + z^10)
sage: C.geometric_genus()
3
Examples of affine curves.
sage: x, y = PolynomialRing(GF(5), 2, 'xy').gens()
sage: C = Curve(y^2 - x^3 - 17*x + y)
sage: C.geometric_genus()
1
sage: C = Curve(y^2 - x^3)
sage: C.geometric_genus()
0
sage: C = Curve(x^10 + y^7 + 1)
sage: C.geometric_genus()
3
"""
try:
return self.__genus
except AttributeError:
self.__genus = self.defining_ideal().genus()
return self.__genus
def union(self, other):
from constructor import Curve
return Curve(AlgebraicScheme_subscheme.union(self, other))
__add__ = union
class Curve_generic_projective(Curve_generic, AlgebraicScheme_subscheme_projective):
pass
