Trac #31892: Conic parametrization broken

The final output here is wrong:
{{{
sage: c = Conic(GF(2), [1,1,1,1,1,0])
....:
sage: c.parametrization()
(Scheme morphism:
   From: Projective Space of dimension 1 over Finite Field of size 2
   To:   Projective Conic Curve over Finite Field of size 2 defined by
x^2 + x*y + y^2 + x*z + y*z
   Defn: Defined on coordinates by sending (x : y) to
         (x*y + y^2 : x^2 + x*y : x^2 + x*y + y^2),
 Scheme morphism:
   From: Projective Conic Curve over Finite Field of size 2 defined by
x^2 + x*y + y^2 + x*z + y*z
   To:   Projective Space of dimension 1 over Finite Field of size 2
   Defn: Defined on coordinates by sending (x : y : z) to
         (y : x))
sage: f, g = c.parametrization()
sage: (g*f).is_one()
False
}}}
The same here:
{{{
sage: R.<x,y,z> = QQ[]
sage: C = Curve(7*x^2 + 2*y*z + z^2)
sage: f, g = C.parametrization(); f,g
(Scheme morphism:
   From: Projective Space of dimension 1 over Rational Field
   To:   Projective Conic Curve over Rational Field defined by 7*x^2 +
2*y*z + z^2
   Defn: Defined on coordinates by sending (x : y) to
         (-2*x*y : x^2 + 7*y^2 : -2*x^2),
 Scheme morphism:
   From: Projective Conic Curve over Rational Field defined by 7*x^2 +
2*y*z + z^2
   To:   Projective Space of dimension 1 over Rational Field
   Defn: Defined on coordinates by sending (x : y : z) to
         (-1/2*x : 1/7*y + 1/14*z))
sage: (g*f).is_one()
False
sage: g([0, -1, 2])
...
ValueError: [0, 0] does not define a valid point since all entries are 0
sage: p = g.domain().defining_polynomial()
sage: p([0, -1, 2])
0
}}}

URL: https://trac.sagemath.org/31892
Reported by: gh-kliem
Ticket author(s): Kwankyu Lee
Reviewer(s): Marco Streng

src/sage/categories/homset.py

@@ -47,8 +47,10 @@
 
 - Simon King (2013-02): added examples
 """
+
 # ****************************************************************************
-#  Copyright (C) 2005 David Kohel <kohel@maths.usyd.edu>, William Stein <wstein@gmail.com>
+#  Copyright (C) 2005 David Kohel <kohel@maths.usyd.edu>,
+#                     William Stein <wstein@gmail.com>
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
 #
@@ -62,7 +64,6 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
-
 from sage.categories.category import Category, JoinCategory
 from . import morphism
 from sage.structure.parent import Parent, Set_generic
@@ -86,7 +87,6 @@ def Hom(X, Y, category=None, check=True):
 
     INPUT:
 
-
     - ``X`` -- an object of a category
 
     - ``Y`` -- an object of a category
@@ -743,8 +743,6 @@ def __bool__(self):
         """
         return True
 
-
-
     def homset_category(self):
         """
         Return the category that this is a Hom in, i.e., this is typically
@@ -1083,7 +1081,7 @@ def __ne__(self, other):
             True
         """
         return not (self == other)
-    
+
     def __contains__(self, x):
         """
         Test whether the parent of the argument is ``self``.

src/sage/schemes/affine/affine_homset.py

@@ -23,15 +23,16 @@
 - Ben Hutz (2018): add numerical point support
 """
 
-
-#*****************************************************************************
-#       Copyright (C) 2006 William Stein <wstein@gmail.com>
+# *****************************************************************************
+#        Copyright (C) 2006 William Stein <wstein@gmail.com>
 #
-#  Distributed under the terms of the GNU General Public License (GPL)
-#  as published by the Free Software Foundation; either version 2 of
-#  the License, or (at your option) any later version.
-#                  http://www.gnu.org/licenses/
-#*****************************************************************************
+#   Distributed under the terms of the GNU General Public License (GPL)
+#   as published by the Free Software Foundation; either version 2 of
+#   the License, or (at your option) any later version.
+#                   http://www.gnu.org/licenses/
+# *****************************************************************************
+
+from copy import copy
 
 from sage.misc.verbose import verbose
 from sage.rings.integer_ring import ZZ
@@ -42,13 +43,13 @@
 from sage.categories.number_fields import NumberFields
 from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
-import sage.schemes.generic.homset
-from copy import copy
+from sage.schemes.generic.homset import SchemeHomset_points, SchemeHomset_generic
+
+# *******************************************************************
+#  Affine varieties
+# *******************************************************************
 
-#*******************************************************************
-# Affine varieties
-#*******************************************************************
-class SchemeHomset_points_spec(sage.schemes.generic.homset.SchemeHomset_generic):
+class SchemeHomset_points_spec(SchemeHomset_generic):
     """
     Set of rational points of an affine variety.
 
@@ -62,7 +63,6 @@ class SchemeHomset_points_spec(sage.schemes.generic.homset.SchemeHomset_generic)
         sage: SchemeHomset_points_spec(Spec(QQ), Spec(QQ))
         Set of rational points of Spectrum of Rational Field
     """
-
     def _element_constructor_(self, *args, **kwds):
         """
         The element constructor.
@@ -86,7 +86,7 @@ def _element_constructor_(self, *args, **kwds):
               Defn: Ring endomorphism of Rational Field
                       Defn: 1 |--> 1
         """
-        return sage.schemes.generic.homset.SchemeHomset_generic._element_constructor_(self, *args, **kwds)
+        return super()._element_constructor_(*args, **kwds)
 
     def _repr_(self):
         """
@@ -101,14 +101,47 @@ def _repr_(self):
             sage: S._repr_()
             'Set of rational points of Spectrum of Rational Field'
         """
-        return 'Set of rational points of '+str(self.codomain())
+        return 'Set of rational points of {}'.format(self.codomain())
+
+
+class SchemeHomset_polynomial_affine_space(SchemeHomset_generic):
+    """
+    Set of morphisms between affine spaces defined by polynomials.
+
+    EXAMPLES::
+
+        sage: A.<x,y> = AffineSpace(2, QQ)
+        sage: Hom(A, A)
+        Set of morphisms
+          From: Affine Space of dimension 2 over Rational Field
+          To:   Affine Space of dimension 2 over Rational Field
+    """
+    def identity(self):
+        """
+        The identity morphism of this homset.
+
+        EXAMPLES::
+
+            sage: A.<x,y> = AffineSpace(2, QQ)
+            sage: I = A.identity_morphism()
+            sage: I.parent()
+            Set of morphisms
+              From: Affine Space of dimension 2 over Rational Field
+              To:   Affine Space of dimension 2 over Rational Field
+            sage: _.identity() == I
+            True
+        """
+        if self.is_endomorphism_set():
+            from sage.schemes.generic.morphism import SchemeMorphism_polynomial_id
+            return SchemeMorphism_polynomial_id(self.domain())
+        raise TypeError("identity map is only defined for endomorphisms")
 
 
+# *******************************************************************
+#  Affine varieties
+# *******************************************************************
 
-#*******************************************************************
-# Affine varieties
-#*******************************************************************
-class SchemeHomset_points_affine(sage.schemes.generic.homset.SchemeHomset_points):
+class SchemeHomset_points_affine(SchemeHomset_points):
     """
     Set of rational points of an affine variety.
 

src/sage/schemes/affine/affine_space.py

@@ -20,21 +20,24 @@
 from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
 from sage.categories.map import Map
 from sage.categories.fields import Fields
-_Fields = Fields()
+from sage.categories.homset import Hom
 from sage.categories.number_fields import NumberFields
 from sage.misc.latex import latex
 from sage.misc.mrange import cartesian_product_iterator
+from sage.matrix.constructor import matrix
 from sage.structure.category_object import normalize_names
 from sage.schemes.generic.scheme import AffineScheme
 from sage.schemes.generic.ambient_space import AmbientSpace
-from sage.schemes.affine.affine_homset import SchemeHomset_points_affine
+from sage.schemes.affine.affine_homset import (SchemeHomset_points_affine,
+                                               SchemeHomset_polynomial_affine_space)
 from sage.schemes.affine.affine_morphism import (SchemeMorphism_polynomial_affine_space,
                                                  SchemeMorphism_polynomial_affine_space_field,
                                                  SchemeMorphism_polynomial_affine_space_finite_field)
 from sage.schemes.affine.affine_point import (SchemeMorphism_point_affine,
                                               SchemeMorphism_point_affine_field,
                                               SchemeMorphism_point_affine_finite_field)
-from sage.matrix.constructor import matrix
+
+_Fields = Fields()
 
 def is_AffineSpace(x):
     r"""
@@ -225,7 +228,6 @@ def __iter__(self):
         for v in cartesian_product_iterator([R for _ in range(n)]):
             yield C._point(AHom, v, check=False)
 
-
     def ngens(self):
         """
         Return the number of generators of self, i.e. the number of
@@ -363,6 +365,20 @@ def _morphism(self, *args, **kwds):
         """
         return SchemeMorphism_polynomial_affine_space(*args, **kwds)
 
+    def _homset(self, *args, **kwds):
+        """
+        Construct the Hom-set.
+
+        EXAMPLES::
+
+            sage: A.<x,y> = AffineSpace(2, QQ)
+            sage: Hom(A, A)
+            Set of morphisms
+              From: Affine Space of dimension 2 over Rational Field
+              To:   Affine Space of dimension 2 over Rational Field
+        """
+        return SchemeHomset_polynomial_affine_space(*args, **kwds)
+
     def _point_homset(self, *args, **kwds):
         """
         Construct a Hom-set for this affine space.

src/sage/schemes/generic/algebraic_scheme.py

@@ -341,6 +341,22 @@ def ambient_space(self):
         """
         return self.__A
 
+    def identity_morphism(self):
+        """
+        Return the identity morphism.
+
+        OUTPUT: the identity morphism of the scheme ``self``
+
+        EXAMPLES::
+
+            sage: X = Spec(QQ)
+            sage: X.identity_morphism()
+            Scheme endomorphism of Spectrum of Rational Field
+              Defn: Identity map
+        """
+        from sage.schemes.generic.morphism import SchemeMorphism_polynomial_id
+        return SchemeMorphism_polynomial_id(self)
+
     def embedding_morphism(self):
         r"""
         Return the default embedding morphism of ``self``.

src/sage/schemes/generic/ambient_space.py

@@ -279,6 +279,27 @@ def defining_polynomials(self):
         """
         return ()
 
+    def identity_morphism(self):
+        """
+        Return the identity morphism.
+
+        OUTPUT: the identity morphism of the scheme ``self``
+
+        EXAMPLES::
+
+            sage: A = AffineSpace(2, GF(3))
+            sage: A.identity_morphism()
+            Scheme endomorphism of Affine Space of dimension 2 over Finite Field of size 3
+              Defn: Identity map
+
+            sage: P = ProjectiveSpace(3, ZZ)
+            sage: P.identity_morphism()
+            Scheme endomorphism of Projective Space of dimension 3 over Integer Ring
+              Defn: Identity map
+        """
+        from sage.schemes.generic.morphism import SchemeMorphism_polynomial_id
+        return SchemeMorphism_polynomial_id(self)
+
     ######################################################################
     # Associated MPolynomial ring generators
     ######################################################################

src/sage/schemes/generic/homset.py

@@ -25,16 +25,15 @@
 - Ben Hutz (June 2012): added support for projective ring
 """
 
-
-#*****************************************************************************
-#       Copyright (C) 2011 Volker Braun <vbraun.name@gmail.com>
-#       Copyright (C) 2006 William Stein <wstein@gmail.com>
+# *****************************************************************************
+#        Copyright (C) 2011 Volker Braun <vbraun.name@gmail.com>
+#        Copyright (C) 2006 William Stein <wstein@gmail.com>
 #
-#  Distributed under the terms of the GNU General Public License (GPL)
-#  as published by the Free Software Foundation; either version 2 of
-#  the License, or (at your option) any later version.
-#                  http://www.gnu.org/licenses/
-#*****************************************************************************
+#   Distributed under the terms of the GNU General Public License (GPL)
+#   as published by the Free Software Foundation; either version 2 of
+#   the License, or (at your option) any later version.
+#                   http://www.gnu.org/licenses/
+# *****************************************************************************
 
 from sage.categories.homset import HomsetWithBase
 from sage.structure.factory import UniqueFactory
@@ -70,9 +69,10 @@ def is_SchemeHomset(H):
     return isinstance(H, SchemeHomset_generic)
 
 
-#*******************************************************************
-# Factory for Hom sets of schemes
-#*******************************************************************
+# *******************************************************************
+#  Factory for Hom sets of schemes
+# *******************************************************************
+
 class SchemeHomsetFactory(UniqueFactory):
     """
     Factory for Hom-sets of schemes.
@@ -212,10 +212,10 @@ def create_object(self, version, key, **extra_args):
 SchemeHomset = SchemeHomsetFactory('sage.schemes.generic.homset.SchemeHomset')
 
 
+# *******************************************************************
+#  Base class
+# *******************************************************************
 
-#*******************************************************************
-# Base class
-#*******************************************************************
 class SchemeHomset_generic(HomsetWithBase):
     r"""
     The base class for Hom-sets of schemes.
@@ -393,9 +393,11 @@ def _element_constructor_(self, x, check=True):
 
         raise TypeError("x must be a ring homomorphism, list or tuple")
 
-#*******************************************************************
-# Base class for points
-#*******************************************************************
+
+# *******************************************************************
+#  Base class for points
+# *******************************************************************
+
 class SchemeHomset_points(SchemeHomset_generic):
     """
     Set of rational points of the scheme.