Python ProjectiveSpace.subscheme方法代碼示例

# 需要導入模塊: from sage.schemes.projective.projective_space import ProjectiveSpace [as 別名]
# 或者: from sage.schemes.projective.projective_space.ProjectiveSpace import subscheme [as 別名]
def EllipticCurve_from_cubic(F, P, morphism=True):
    r"""
    Construct an elliptic curve from a ternary cubic with a rational point.

    If you just want the Weierstrass form and are not interested in
    the morphism then it is easier to use
    :func:`~sage.schemes.elliptic_curves.jacobian.Jacobian`
    instead. This will construct the same elliptic curve but you don't
    have to supply the point ``P``.

    INPUT:

    - ``F`` -- a homogeneous cubic in three variables with rational
      coefficients, as a polynomial ring element, defining a smooth
      plane cubic curve.

    - ``P`` -- a 3-tuple `(x,y,z)` defining a projective point on the
      curve `F=0`. Need not be a flex, but see caveat on output.

    - ``morphism`` -- boolean (default: ``True``). Whether to return
      the morphism or just the elliptic curve.

    OUTPUT:

    An elliptic curve in long Weierstrass form isomorphic to the curve
    `F=0`.

    If ``morphism=True`` is passed, then a birational equivalence
    between F and the Weierstrass curve is returned. If the point
    happens to be a flex, then this is an isomorphism.

    EXAMPLES:

    First we find that the Fermat cubic is isomorphic to the curve
    with Cremona label 27a1::

        sage: R.<x,y,z> = QQ[]
        sage: cubic = x^3+y^3+z^3
        sage: P = [1,-1,0]
        sage: E = EllipticCurve_from_cubic(cubic, P, morphism=False); E
        Elliptic Curve defined by y^2 + 2*x*y + 1/3*y = x^3 - x^2 - 1/3*x - 1/27 over Rational Field
        sage: E.cremona_label()
        '27a1'
        sage: EllipticCurve_from_cubic(cubic, [0,1,-1], morphism=False).cremona_label()
        '27a1'
        sage: EllipticCurve_from_cubic(cubic, [1,0,-1], morphism=False).cremona_label()
        '27a1'

    Next we find the minimal model and conductor of the Jacobian of the
    Selmer curve::

        sage: R.<a,b,c> = QQ[]
        sage: cubic = a^3+b^3+60*c^3
        sage: P = [1,-1,0]
        sage: E = EllipticCurve_from_cubic(cubic, P, morphism=False);  E
        Elliptic Curve defined by y^2 + 2*x*y + 20*y = x^3 - x^2 - 20*x - 400/3 over Rational Field
        sage: E.minimal_model()
        Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field
        sage: E.conductor()
        24300

    We can also get the birational equivalence to and from the
    Weierstrass form. We start with an example where ``P`` is a flex
    and the equivalence is an isomorphism::

        sage: f = EllipticCurve_from_cubic(cubic, P, morphism=True)
        sage: f
        Scheme morphism:
          From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
                a^3 + b^3 + 60*c^3
          To:   Elliptic Curve defined by y^2 + 2*x*y + 20*y = x^3 - x^2 - 20*x - 400/3
                over Rational Field
          Defn: Defined on coordinates by sending (a : b : c) to
                (-c : -b + c : 1/20*a + 1/20*b)

        sage: finv = f.inverse();  finv
        Scheme morphism:
          From: Elliptic Curve defined by y^2 + 2*x*y + 20*y = x^3 - x^2 - 20*x - 400/3
                over Rational Field
          To:   Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
          a^3 + b^3 + 60*c^3
          Defn: Defined on coordinates by sending (x : y : z) to
                (x + y + 20*z : -x - y : -x)

    We verify that `f` maps the chosen point `P=(1,-1,0)` on the cubic
    to the origin of the elliptic curve::

        sage: f([1,-1,0])
        (0 : 1 : 0)
        sage: finv([0,1,0])
        (-1 : 1 : 0)

    To verify the output, we plug in the polynomials to check that
    this indeed transforms the cubic into Weierstrass form::

        sage: cubic(finv.defining_polynomials()) * finv.post_rescaling()
        -x^3 + x^2*z + 2*x*y*z + y^2*z + 20*x*z^2 + 20*y*z^2 + 400/3*z^3

        sage: E.defining_polynomial()(f.defining_polynomials()) * f.post_rescaling()
        a^3 + b^3 + 60*c^3
#.........這裏部分代碼省略.........

# 需要導入模塊: from sage.schemes.projective.projective_space import ProjectiveSpace [as 別名]
# 或者: from sage.schemes.projective.projective_space.ProjectiveSpace import subscheme [as 別名]
    def segre_embedding(self, PP=None, var='u'):
        r"""
        Return the Segre embedding of this space into the appropriate
        projective space.

        INPUT:

        -  ``PP`` -- (default: ``None``) ambient image projective space;
            this is constructed if it is not given.

        - ``var`` -- string, variable name of the image projective space, default `u` (optional).

        OUTPUT:

        Hom -- from this space to the appropriate subscheme of projective space.

        .. TODO::

            Cartesian products with more than two components.

        EXAMPLES::

            sage: X.<y0,y1,y2,y3,y4,y5> = ProductProjectiveSpaces(ZZ, [2, 2])
            sage: phi = X.segre_embedding(); phi
            Scheme morphism:
              From: Product of projective spaces P^2 x P^2 over Integer Ring
              To:   Closed subscheme of Projective Space of dimension 8 over Integer Ring defined by:
              -u5*u7 + u4*u8,
              -u5*u6 + u3*u8,
              -u4*u6 + u3*u7,
              -u2*u7 + u1*u8,
              -u2*u4 + u1*u5,
              -u2*u6 + u0*u8,
              -u1*u6 + u0*u7,
              -u2*u3 + u0*u5,
              -u1*u3 + u0*u4
              Defn: Defined by sending (y0 : y1 : y2 , y3 : y4 : y5) to
                    (y0*y3 : y0*y4 : y0*y5 : y1*y3 : y1*y4 : y1*y5 : y2*y3 : y2*y4 : y2*y5).

            ::

            sage: T = ProductProjectiveSpaces([1, 2], CC, 'z')
            sage: T.segre_embedding()
            Scheme morphism:
              From: Product of projective spaces P^1 x P^2 over Complex Field with 53 bits of precision
              To:   Closed subscheme of Projective Space of dimension 5 over Complex Field with 53 bits of precision defined by:
              -u2*u4 + u1*u5,
              -u2*u3 + u0*u5,
              -u1*u3 + u0*u4
              Defn: Defined by sending (z0 : z1 , z2 : z3 : z4) to
                    (z0*z2 : z0*z3 : z0*z4 : z1*z2 : z1*z3 : z1*z4).

            ::

            sage: T = ProductProjectiveSpaces([1, 2, 1], QQ, 'z')
            sage: T.segre_embedding()
            Scheme morphism:
              From: Product of projective spaces P^1 x P^2 x P^1 over Rational Field
              To:   Closed subscheme of Projective Space of dimension 11 over
            Rational Field defined by:
              -u9*u10 + u8*u11,
              -u7*u10 + u6*u11,
              -u7*u8 + u6*u9,
              -u5*u10 + u4*u11,
              -u5*u8 + u4*u9,
              -u5*u6 + u4*u7,
              -u5*u9 + u3*u11,
              -u5*u8 + u3*u10,
              -u5*u8 + u2*u11,
              -u4*u8 + u2*u10,
              -u3*u8 + u2*u9,
              -u3*u6 + u2*u7,
              -u3*u4 + u2*u5,
              -u5*u7 + u1*u11,
              -u5*u6 + u1*u10,
              -u3*u7 + u1*u9,
              -u3*u6 + u1*u8,
              -u5*u6 + u0*u11,
              -u4*u6 + u0*u10,
              -u3*u6 + u0*u9,
              -u2*u6 + u0*u8,
              -u1*u6 + u0*u7,
              -u1*u4 + u0*u5,
              -u1*u2 + u0*u3
              Defn: Defined by sending (z0 : z1 , z2 : z3 : z4 , z5 : z6) to
                    (z0*z2*z5 : z0*z2*z6 : z0*z3*z5 : z0*z3*z6 : z0*z4*z5 : z0*z4*z6
            : z1*z2*z5 : z1*z2*z6 : z1*z3*z5 : z1*z3*z6 : z1*z4*z5 : z1*z4*z6).
        """
        N = self._dims
        M = prod([n+1 for n in N]) - 1
        CR = self.coordinate_ring()

        vars = list(self.coordinate_ring().variable_names()) + [var + str(i) for i in range(M+1)]
        R = PolynomialRing(self.base_ring(), self.ngens()+M+1, vars, order='lex')

        #set-up the elimination for the segre embedding
        mapping = []
        k = self.ngens()
        index = self.num_components()*[0]
        for count in range(M + 1):
#.........這裏部分代碼省略.........

# 需要導入模塊: from sage.schemes.projective.projective_space import ProjectiveSpace [as 別名]
# 或者: from sage.schemes.projective.projective_space.ProjectiveSpace import subscheme [as 別名]
    def segre_embedding(self, PP=None, var='u'):
        r"""
        Return the Segre embedding of ``self`` into the appropriate
        projective space.

        INPUT:

        -  ``PP`` -- (default: ``None``) ambient image projective space;
            this is constructed if it is not given.

        - ``var`` -- string, variable name of the image projective space, default `u` (optional)

        OUTPUT:

        Hom -- from ``self`` to the appropriate subscheme of projective space

        .. TODO::

            Cartesian products with more than two components

        EXAMPLES::

            sage: X.<y0,y1,y2,y3,y4,y5> = ProductProjectiveSpaces(ZZ,[2,2])
            sage: phi = X.segre_embedding(); phi
            Scheme morphism:
              From: Product of projective spaces P^2 x P^2 over Integer Ring
              To:   Closed subscheme of Projective Space of dimension 8 over Integer Ring defined by:
              -u5*u7 + u4*u8,
              -u5*u6 + u3*u8,
              -u4*u6 + u3*u7,
              -u2*u7 + u1*u8,
              -u2*u4 + u1*u5,
              -u2*u6 + u0*u8,
              -u1*u6 + u0*u7,
              -u2*u3 + u0*u5,
              -u1*u3 + u0*u4
              Defn: Defined by sending (y0 : y1 : y2 , y3 : y4 : y5) to
                    (y0*y3 : y0*y4 : y0*y5 : y1*y3 : y1*y4 : y1*y5 : y2*y3 : y2*y4 : y2*y5).

            ::

            sage: T = ProductProjectiveSpaces([1,2],CC,'z')
            sage: T.segre_embedding()
            Scheme morphism:
              From: Product of projective spaces P^1 x P^2 over Complex Field with 53 bits of precision
              To:   Closed subscheme of Projective Space of dimension 5 over Complex Field with 53 bits of precision defined by:
              -u2*u4 + u1*u5,
              -u2*u3 + u0*u5,
              -u1*u3 + u0*u4
              Defn: Defined by sending (z0 : z1 , z2 : z3 : z4) to
                    (z0*z2 : z0*z3 : z0*z4 : z1*z2 : z1*z3 : z1*z4).
        """
        N = self._dims
        if len(N) > 2:
            raise NotImplementedError("Cannot have more than two components.")
        M = (N[0]+1)*(N[1]+1)-1

        vars = list(self.coordinate_ring().variable_names()) + [var + str(i) for i in range(M+1)]
        R = PolynomialRing(self.base_ring(),self.ngens()+M+1, vars, order='lex')

        #set-up the elimination for the segre embedding
        mapping = []
        k = self.ngens()
        for i in range(N[0]+1):
            for j in range(N[0]+1,N[0]+N[1]+2):
                mapping.append(R.gen(k)-R(self.gen(i)*self.gen(j)))
                k+=1

        #change the defining ideal of the subscheme into the variables
        I = R.ideal(list(self.defining_polynomials()) + mapping)
        J = I.groebner_basis()
        s = set(R.gens()[:self.ngens()])
        n = len(J)-1
        L = []
        while s.isdisjoint(J[n].variables()):
            L.append(J[n])
            n = n-1

        #create new subscheme
        if PP is None:
            PS = ProjectiveSpace(self.base_ring(),M,R.gens()[self.ngens():])
            Y = PS.subscheme(L)
        else:
            if PP.dimension_relative()!= M:
                raise ValueError("Projective Space %s must be dimension %s")%(PP, M)
            S = PP.coordinate_ring()
            psi = R.hom([0]*(N[0]+N[1]+2) + list(S.gens()),S)
            L = [psi(l) for l in L]
            Y = PP.subscheme(L)

        #create embedding for points
        mapping = []
        for i in range(N[0]+1):
            for j in range(N[0]+1,N[0]+N[1]+2):
                mapping.append(self.gen(i)*self.gen(j))
        phi = self.hom(mapping,Y)

        return phi

# 需要導入模塊: from sage.schemes.projective.projective_space import ProjectiveSpace [as 別名]
# 或者: from sage.schemes.projective.projective_space.ProjectiveSpace import subscheme [as 別名]
    def dual(self):
        r"""
        Return the projective dual of the given subscheme of projective space.

        INPUT:

        - ``X`` -- A subscheme of projective space. At present, ``X`` is
          required to be an irreducible and reduced hypersurface defined
          over `\QQ` or a finite field.

        OUTPUT:

        - The dual of ``X`` as a subscheme of the dual projective space.

        EXAMPLES:

        The dual of a smooth conic in the plane is also a smooth conic::

            sage: R.<x, y, z> = QQ[]
            sage: P.<x, y, z> = ProjectiveSpace(2, QQ)
            sage: I = R.ideal(x^2 + y^2 + z^2)
            sage: X = P.subscheme(I)
            sage: X.dual()
            Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
              y0^2 + y1^2 + y2^2

        The dual of the twisted cubic curve in projective 3-space is a singular
        quartic surface. In the following example, we compute the dual of this
        surface, which by double duality is equal to the twisted cubic itself.
        The output is the twisted cubic as an intersection of three quadrics::

            sage: R.<x, y, z, w> = QQ[]
            sage: P.<x, y, z, w> = ProjectiveSpace(3, QQ)
            sage: I = R.ideal(y^2*z^2 - 4*x*z^3 - 4*y^3*w + 18*x*y*z*w - 27*x^2*w^2)
            sage: X = P.subscheme(I)
            sage: X.dual()
            Closed subscheme of Projective Space of dimension 3 over
            Rational Field defined by:
              y2^2 - y1*y3,
              y1*y2 - y0*y3,
              y1^2 - y0*y2

        The singular locus of the quartic surface in the last example
        is itself supported on a twisted cubic::

            sage: X.Jacobian().radical()
            Ideal (z^2 - 3*y*w, y*z - 9*x*w, y^2 - 3*x*z) of Multivariate
            Polynomial Ring in x, y, z, w over Rational Field

        An example over a finite field::

            sage: R = PolynomialRing(GF(61), 'a,b,c')
            sage: P.<a, b, c> = ProjectiveSpace(2, R.base_ring())
            sage: X = P.subscheme(R.ideal(a*a+2*b*b+3*c*c))
            sage: X.dual()
            Closed subscheme of Projective Space of dimension 2 over
            Finite Field of size 61 defined by:
            y0^2 - 30*y1^2 - 20*y2^2

        TESTS::

            sage: R = PolynomialRing(Qp(3), 'a,b,c')
            sage: P.<a, b, c> = ProjectiveSpace(2, R.base_ring())
            sage: X = P.subscheme(R.ideal(a*a+2*b*b+3*c*c))
            sage: X.dual()
            Traceback (most recent call last):
            ...
            NotImplementedError: base ring must be QQ or a finite field
        """
        from sage.libs.singular.function_factory import ff

        K = self.base_ring()
        if not(is_RationalField(K) or is_FiniteField(K)):
            raise NotImplementedError("base ring must be QQ or a finite field")
        I = self.defining_ideal()
        m = I.ngens()
        n = I.ring().ngens() - 1
        if (m != 1 or (n < 1) or I.is_zero()
            or I.is_trivial() or not I.is_prime()):
            raise NotImplementedError("At the present, the method is only"
                                      " implemented for irreducible and"
                                      " reduced hypersurfaces and the given"
                                      " list of generators for the ideal must"
                                      " have exactly one element.")
        R = PolynomialRing(K, 'x', n + 1)
        from sage.schemes.projective.projective_space import ProjectiveSpace
        Pd = ProjectiveSpace(n, K, 'y')
        Rd = Pd.coordinate_ring()
        x = R.variable_names()
        y = Rd.variable_names()
        S = PolynomialRing(K, x + y + ('t',))
        if S.has_coerce_map_from(I.ring()):
            T = PolynomialRing(K, 'w', n + 1)
            I_S = (I.change_ring(T)).change_ring(S)
        else:
            I_S = I.change_ring(S)
        f_S = I_S.gens()[0]
        z = S.gens()
        J = I_S
        for i in range(n + 1):
#.........這裏部分代碼省略.........