20676: changes from review, and spacing fixes.

src/sage/schemes/curves/affine_curve.py

@@ -91,14 +91,17 @@ def __init__(self, A, X):
         if d != 1:
             raise ValueError("defining equations (=%s) define a scheme of dimension %s != 1"%(X,d))
 
-    def projective_closure(self, i=0):
+    def projective_closure(self, i=0, PP=None):
         r"""
         Return the projective closure of this affine curve.
 
         INPUT:
 
-        - ``i`` - (default: 0) the index of the affine coordinate chart of the projective space that the affine ambient space
-          of this curve embeds into.
+        - ``i`` -- (default: 0) the index of the affine coordinate chart of the projective space that the affine
+          ambient space of this curve embeds into.
+
+        - ``PP`` -- (default: None) ambient projective space to compute the projective closure in. This is
+          constructed if it is not given.
 
         OUTPUT:
 
@@ -127,9 +130,17 @@ def projective_closure(self, i=0):
             sage: C.projective_closure(1)
             Projective Plane Curve over Complex Field with 53 bits of precision defined by
             x0^3 - x0*x1^2 + x1^3 - x1^2*x2
+
+        ::
+
+            sage: A.<x,y> = AffineSpace(QQ, 2)
+            sage: P.<u,v,w> = ProjectiveSpace(QQ, 2)
+            sage: C = Curve([y - x^2], A)
+            sage: C.projective_closure(1, P).ambient_space() == P
+            True
         """
         from constructor import Curve
-        return Curve(AlgebraicScheme_subscheme_affine.projective_closure(self, i))
+        return Curve(AlgebraicScheme_subscheme_affine.projective_closure(self, i, PP))
 
 class AffinePlaneCurve(AffineCurve):
     def __init__(self, A, f):

src/sage/schemes/curves/projective_curve.py

@@ -87,15 +87,17 @@ def __init__(self, A, X):
         if d != 1:
             raise ValueError("defining equations (=%s) define a scheme of dimension %s != 1"%(X,d))
 
-    def affine_patch(self, i):
+    def affine_patch(self, i, AA=None):
         r"""
         Return the i-th affine patch of this projective curve.
 
         INPUT:
 
-        - ``i`` - affine coordinate chart of the projective ambient space of this curve to compute affine patch
+        - ``i`` -- affine coordinate chart of the projective ambient space of this curve to compute affine patch
           with respect to.
 
+        - ``AA`` -- (default: None) ambient affine space, this is constructed if it is not given.
+
         OUTPUT:
 
         - a curve in affine space.
@@ -114,9 +116,17 @@ def affine_patch(self, i):
             sage: C = Curve(x^3 - x^2*y + y^3 - x^2*z, P)
             sage: C.affine_patch(1)
             Affine Plane Curve over Rational Field defined by x0^3 - x0^2*x1 - x0^2 + 1
+
+        ::
+
+            sage: A.<x,y> = AffineSpace(QQ, 2)
+            sage: P.<u,v,w> = ProjectiveSpace(QQ, 2)
+            sage: C = Curve([u^2 - v^2], P)
+            sage: C.affine_patch(1, A).ambient_space() == A
+            True
         """
         from constructor import Curve
-        return Curve(AlgebraicScheme_subscheme_projective.affine_patch(self, i))
+        return Curve(AlgebraicScheme_subscheme_projective.affine_patch(self, i, AA))
 
 class ProjectivePlaneCurve(ProjectiveCurve):
     def __init__(self, A, f):

src/sage/schemes/generic/algebraic_scheme.py

@@ -1825,6 +1825,9 @@ def projective_embedding(self, i=None, PP=None):
         Returns a morphism from this affine scheme into an ambient
         projective space of the same dimension.
 
+        The codomain of this morphism is the projective closure of this affine scheme in ``PP``,
+        if given, or otherwise in a new projective space that is constructed.
+
         INPUT:
 
         -  ``i`` -- integer (default: dimension of self = last
@@ -1912,8 +1915,8 @@ def projective_embedding(self, i=None, PP=None):
         # Groebner basis w.r.t. a graded monomial order computed here to ensure
         # after homogenization, the basis elements will generate the defining
         # ideal of the projective closure of this affine subscheme
-        R = AA.coordinate_ring().change_ring(order='degrevlex')
-        G = R.ideal([R(f) for f in self.defining_polynomials()]).groebner_basis()
+        R = AA.coordinate_ring()
+        G = self.defining_ideal().groebner_basis()
         v = list(PP.gens())
         z = v.pop(i)
         phi = R.hom(v,PR)
@@ -1925,7 +1928,7 @@ def projective_embedding(self, i=None, PP=None):
         self.__projective_embedding[i] = phi
         return phi
 
-    def projective_closure(self,i=None):
+    def projective_closure(self, i=None, PP=None):
         r"""
         Return the projective closure of this affine subscheme.
 
@@ -1935,6 +1938,9 @@ def projective_closure(self,i=None):
           closure of this affine subscheme. The embedding used is the one which has a 1 in the
           i-th coordinate, numbered from 0.
 
+        -  ``PP`` -- (default: None) ambient projective space, i.e., ambient space
+           of codomain of morphism; this is constructed if it is not given.
+
         OUTPUT:
 
         - a projective subscheme.
@@ -1952,8 +1958,16 @@ def projective_closure(self,i=None):
               x0*x2 - x3*x4,
               x1*x2 - x0*x3,
               x2^2 - x1*x3
+
+        ::
+
+            sage: A.<x,y,z> = AffineSpace(QQ, 3)
+            sage: P.<a,b,c,d> = ProjectiveSpace(QQ, 3)
+            sage: X = A.subscheme([z - x^2 - y^2])
+            sage: X.projective_closure(1, P).ambient_space() == P
+            True
         """
-        return self.projective_embedding(i).codomain()
+        return self.projective_embedding(i, PP).codomain()
 
     def is_smooth(self, point=None):
         r"""