CoCalc -- projective

GitHub Repository: sagemath/sagesmc
Path: blob/master/src/sage/schemes/projective/projective_homset.py
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r"""
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Set of homomorphisms between two proejctive schemes
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For schemes `X` and `Y`, this module implements the set of morphisms
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`Hom(X,Y)`. This is done by :class:`SchemeHomset_generic`.
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As a special case, the Hom-sets can also represent the points of a
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scheme. Recall that the `K`-rational points of a scheme `X` over `k`
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can be identified with the set of morphisms `Spec(K) \to X`. In Sage
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the rational points are implemented by such scheme morphisms. This is
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done by :class:`SchemeHomset_points` and its subclasses.
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.. note::
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    You should not create the Hom-sets manually. Instead, use the
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    :meth:`~sage.structure.parent.Hom` method that is inherited by all
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    schemes.
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AUTHORS:
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- William Stein (2006): initial version.
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- Volker Braun (2011-08-11): significant improvement and refactoring.
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- Ben Hutz (June 2012): added support for projective ring
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"""
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#*****************************************************************************
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#       Copyright (C) 2011 Volker Braun <[email protected]>
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#       Copyright (C) 2006 William Stein <[email protected]>
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#
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#  Distributed under the terms of the GNU General Public License (GPL)
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#  as published by the Free Software Foundation; either version 2 of
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#  the License, or (at your option) any later version.
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#                  http://www.gnu.org/licenses/
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#*****************************************************************************
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from sage.rings.all import ZZ
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from sage.schemes.generic.homset import SchemeHomset_points
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from sage.rings.rational_field import is_RationalField
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from sage.rings.finite_rings.constructor import is_FiniteField
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#*******************************************************************
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# Projective varieties
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#*******************************************************************
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class SchemeHomset_points_projective_field(SchemeHomset_points):
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    """
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    Set of rational points of a projective variety over a field.
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    INPUT:
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    See :class:`SchemeHomset_generic`.
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    EXAMPLES::
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        sage: from sage.schemes.projective.projective_homset import SchemeHomset_points_projective_field
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        sage: SchemeHomset_points_projective_field(Spec(QQ), ProjectiveSpace(QQ,2))
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        Set of rational points of Projective Space of dimension 2 over Rational Field
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    """
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    def points(self, B=0):
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        """
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        Return some or all rational points of a projective scheme.
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        INPUT:
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        - `B` -- integer (optional, default=0). The bound for the
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          coordinates.
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        OUTPUT:
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        A list of points. Over a finite field, all points are
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        returned. Over an infinite field, all points satisfying the
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        bound are returned.
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        EXAMPLES::
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            sage: P1 = ProjectiveSpace(GF(2),1)
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            sage: F.<a> = GF(4,'a')
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            sage: P1(F).points()
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            [(0 : 1), (1 : 0), (1 : 1), (a : 1), (a + 1 : 1)]
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        """
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        from sage.schemes.projective.projective_rational_point import enum_projective_rational_field
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        from sage.schemes.projective.projective_rational_point import enum_projective_finite_field
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        R = self.value_ring()
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        if is_RationalField(R):
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            if not B > 0:
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                raise TypeError, "A positive bound B (= %s) must be specified."%B
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            return enum_projective_rational_field(self,B)
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        elif is_FiniteField(R):
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            return enum_projective_finite_field(self.extended_codomain())
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        else:
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            raise TypeError, "Unable to enumerate points over %s."%R
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class SchemeHomset_points_projective_ring(SchemeHomset_points):
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    """
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    Set of rational points of a projective variety over a commutative ring.
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    INPUT:
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    See :class:`SchemeHomset_generic`.
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    EXAMPLES::
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        sage: from sage.schemes.projective.projective_homset import SchemeHomset_points_projective_ring
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        sage: SchemeHomset_points_projective_ring(Spec(ZZ), ProjectiveSpace(ZZ,2))
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        Set of rational points of Projective Space of dimension 2 over Integer Ring
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    """
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    def points(self, B=0):
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        """
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        Return some or all rational points of a projective scheme.
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        INPUT:
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        - `B` -- integer (optional, default=0). The bound for the
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          coordinates.
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        EXAMPLES::
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            sage: from sage.schemes.projective.projective_homset import SchemeHomset_points_projective_ring
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            sage: H = SchemeHomset_points_projective_ring(Spec(ZZ), ProjectiveSpace(ZZ,2))
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            sage: H.points(3)
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            [(0 : 0 : 1), (0 : 1 : -3), (0 : 1 : -2), (0 : 1 : -1), (0 : 1 : 0), (0
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            : 1 : 1), (0 : 1 : 2), (0 : 1 : 3), (0 : 2 : -3), (0 : 2 : -1), (0 : 2 :
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            1), (0 : 2 : 3), (0 : 3 : -2), (0 : 3 : -1), (0 : 3 : 1), (0 : 3 : 2),
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            (1 : -3 : -3), (1 : -3 : -2), (1 : -3 : -1), (1 : -3 : 0), (1 : -3 : 1),
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            (1 : -3 : 2), (1 : -3 : 3), (1 : -2 : -3), (1 : -2 : -2), (1 : -2 : -1),
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            (1 : -2 : 0), (1 : -2 : 1), (1 : -2 : 2), (1 : -2 : 3), (1 : -1 : -3),
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            (1 : -1 : -2), (1 : -1 : -1), (1 : -1 : 0), (1 : -1 : 1), (1 : -1 : 2),
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            (1 : -1 : 3), (1 : 0 : -3), (1 : 0 : -2), (1 : 0 : -1), (1 : 0 : 0), (1
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            : 0 : 1), (1 : 0 : 2), (1 : 0 : 3), (1 : 1 : -3), (1 : 1 : -2), (1 : 1 :
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            -1), (1 : 1 : 0), (1 : 1 : 1), (1 : 1 : 2), (1 : 1 : 3), (1 : 2 : -3),
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            (1 : 2 : -2), (1 : 2 : -1), (1 : 2 : 0), (1 : 2 : 1), (1 : 2 : 2), (1 :
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            2 : 3), (1 : 3 : -3), (1 : 3 : -2), (1 : 3 : -1), (1 : 3 : 0), (1 : 3 :
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            1), (1 : 3 : 2), (1 : 3 : 3), (2 : -3 : -3), (2 : -3 : -2), (2 : -3 :
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            -1), (2 : -3 : 0), (2 : -3 : 1), (2 : -3 : 2), (2 : -3 : 3), (2 : -2 :
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            -3), (2 : -2 : -1), (2 : -2 : 1), (2 : -2 : 3), (2 : -1 : -3), (2 : -1 :
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            -2), (2 : -1 : -1), (2 : -1 : 0), (2 : -1 : 1), (2 : -1 : 2), (2 : -1 :
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            3), (2 : 0 : -3), (2 : 0 : -1), (2 : 0 : 1), (2 : 0 : 3), (2 : 1 : -3),
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            (2 : 1 : -2), (2 : 1 : -1), (2 : 1 : 0), (2 : 1 : 1), (2 : 1 : 2), (2 :
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            1 : 3), (2 : 2 : -3), (2 : 2 : -1), (2 : 2 : 1), (2 : 2 : 3), (2 : 3 :
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            -3), (2 : 3 : -2), (2 : 3 : -1), (2 : 3 : 0), (2 : 3 : 1), (2 : 3 : 2),
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            (2 : 3 : 3), (3 : -3 : -2), (3 : -3 : -1), (3 : -3 : 1), (3 : -3 : 2),
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            (3 : -2 : -3), (3 : -2 : -2), (3 : -2 : -1), (3 : -2 : 0), (3 : -2 : 1),
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            (3 : -2 : 2), (3 : -2 : 3), (3 : -1 : -3), (3 : -1 : -2), (3 : -1 : -1),
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            (3 : -1 : 0), (3 : -1 : 1), (3 : -1 : 2), (3 : -1 : 3), (3 : 0 : -2), (3
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            : 0 : -1), (3 : 0 : 1), (3 : 0 : 2), (3 : 1 : -3), (3 : 1 : -2), (3 : 1
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            : -1), (3 : 1 : 0), (3 : 1 : 1), (3 : 1 : 2), (3 : 1 : 3), (3 : 2 : -3),
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            (3 : 2 : -2), (3 : 2 : -1), (3 : 2 : 0), (3 : 2 : 1), (3 : 2 : 2), (3 :
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            2 : 3), (3 : 3 : -2), (3 : 3 : -1), (3 : 3 : 1), (3 : 3 : 2)]
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        """
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        R = self.value_ring()
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        if R == ZZ:
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            if not B > 0:
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                raise TypeError, "A positive bound B (= %s) must be specified."%B
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            from sage.schemes.projective.projective_rational_point import enum_projective_rational_field
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            return enum_projective_rational_field(self,B)
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        else:
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            raise TypeError, "Unable to enumerate points over %s."%R
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#*******************************************************************
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# Abelian varieties
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#*******************************************************************
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class SchemeHomset_points_abelian_variety_field(SchemeHomset_points_projective_field):
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    r"""
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    Set of rational points of an abelian variety.
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    INPUT:
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    See :class:`SchemeHomset_generic`.
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    TESTS:
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    The bug reported at trac #1785 is fixed::
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        sage: K.<a> = NumberField(x^2 + x - (3^3-3))
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        sage: E = EllipticCurve('37a')
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        sage: X = E(K)
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        sage: X
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        Abelian group of points on Elliptic Curve defined by
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        y^2 + y = x^3 + (-1)*x over Number Field in a with
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        defining polynomial x^2 + x - 24
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        sage: P = X([3,a])
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        sage: P
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        (3 : a : 1)
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        sage: P in E
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        False
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        sage: P in E.base_extend(K)
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        True
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        sage: P in X.codomain()
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        False
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        sage: P in X.extended_codomain()
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        True
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    """
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    def _element_constructor_(self, *v, **kwds):
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        """
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        The element contstructor.
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        INPUT:
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        - ``v`` -- anything that determines a scheme morphism in the
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          Hom-set.
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        OUTPUT:
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        The scheme morphism determined by ``v``.
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        EXAMPLES::
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            sage: E = EllipticCurve('37a')
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            sage: X = E(QQ)
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            sage: P = X([0,1,0]);  P
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            (0 : 1 : 0)
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            sage: type(P)
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            <class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_number_field'>
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        TESTS::
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            sage: X._element_constructor_([0,1,0])
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            (0 : 1 : 0)
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        """
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        if len(v) == 1:
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            v = v[0]
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        return self.codomain()._point(self.extended_codomain(), v, **kwds)
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    def _repr_(self):
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        """
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        Return a string representation of ``self``.
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        OUTPUT:
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        String.
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        EXAMPLES::
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            sage: E = EllipticCurve('37a')
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            sage: X = E(QQ)
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            sage: X._repr_()
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            'Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field'
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        """
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        s = 'Abelian group of points on ' + str(self.extended_codomain())
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        return s
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    def base_extend(self, R):
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        """
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        Extend the base ring.
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        This is currently not implemented except for the trivial case
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        ``R==ZZ``.
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        INPUT:
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        - ``R`` -- a ring.
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        EXAMPLES::
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            sage: E = EllipticCurve('37a')
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            sage: Hom = E.point_homset();  Hom
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            Abelian group of points on Elliptic Curve defined
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            by y^2 + y = x^3 - x over Rational Field
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            sage: Hom.base_ring()
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            Integer Ring
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            sage: Hom.base_extend(QQ)
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            Traceback (most recent call last):
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            ...
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            NotImplementedError: Abelian variety point sets are not
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            implemented as modules over rings other than ZZ.
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        """
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        if R is not ZZ:
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            raise NotImplementedError('Abelian variety point sets are not '
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                            'implemented as modules over rings other than ZZ.')
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        return self
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from sage.structure.sage_object import register_unpickle_override
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register_unpickle_override('sage.schemes.generic.homset',
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                           'SchemeHomsetModule_abelian_variety_coordinates_field',
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                           SchemeHomset_points_abelian_variety_field)
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