constructor.py

"""
Plane curve constructors

AUTHORS:

- William Stein (2005-11-13)

- David Kohel (2006-01)
"""

#*****************************************************************************
#       Copyright (C) 2005 William Stein <[email protected]>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#
#    This code is distributed in the hope that it will be useful,
#    but WITHOUT ANY WARRANTY; without even the implied warranty of
#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
#    General Public License for more details.
#
#  The full text of the GPL is available at:
#
#                  http://www.gnu.org/licenses/
#*****************************************************************************

from sage.rings.polynomial.multi_polynomial_element import is_MPolynomial
from sage.rings.polynomial.multi_polynomial_ring import is_MPolynomialRing
from sage.rings.finite_rings.constructor import is_FiniteField

from sage.structure.all import Sequence

from sage.schemes.generic.all import (is_AmbientSpace, is_AlgebraicScheme)

from sage.schemes.affine.all import AffineSpace

from sage.schemes.projective.all import ProjectiveSpace


from projective_curve import (ProjectiveCurve_generic,
                              ProjectiveSpaceCurve_generic,
                              ProjectiveCurve_finite_field,
                              ProjectiveCurve_prime_finite_field)

from affine_curve import (AffineCurve_generic,
                          AffineSpaceCurve_generic,
                          AffineCurve_finite_field,
                          AffineCurve_prime_finite_field)

from sage.schemes.plane_conics.constructor import Conic

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