gh-40917: switch to n_components in schemes
    
and keep the old "num_components" as an alias

cf #40887 and #40914

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
    
URL: #40917
Reported by: Frédéric Chapoton
Reviewer(s): Martin Rubey

Author: Release Manager

src/sage/schemes/product_projective/morphism.py

@@ -285,7 +285,7 @@ def __eq__(self, right):
             return False
         PP = self.parent().codomain()
 
-        n = PP.num_components()
+        n = PP.n_components()
         dim = [ P.ngens() for P in PP ]
         dim_prefix = [0,dim[0]]
 
@@ -332,7 +332,7 @@ def __ne__(self, right):
             return True
         PP = self.parent().codomain()
 
-        n = PP.num_components()
+        n = PP.n_components()
         dim = [ P.ngens() for P in PP ]
         dim_prefix = [0,dim[0]]
 
@@ -393,7 +393,7 @@ def is_morphism(self):
             f = self.change_ring(T.base_ring().fraction_field())
             T = T.change_ring(T.base_ring().fraction_field())
 
-        for i in range(S.num_components()):
+        for i in range(S.n_components()):
             t = S[i].dimension_relative() + 1
             X = T.subscheme(list(f)[m : m + t])
             if X.dimension() > -1:

src/sage/schemes/product_projective/point.py

@@ -215,7 +215,7 @@ def __copy__(self):
             sage: P == Q
             True
         """
-        P = [copy(self[i]) for i in range(self.codomain().ambient_space().num_components())]
+        P = [copy(self[i]) for i in range(self.codomain().ambient_space().n_components())]
         return (self.codomain().point(P, False))
 
     def __iter__(self):
@@ -250,7 +250,7 @@ def __len__(self):
             4
         """
         image = self.codomain().ambient_space()
-        return image.dimension() + image.num_components()
+        return image.dimension() + image.n_components()
 
     def __hash__(self):
         """
@@ -308,7 +308,7 @@ def normalize_coordinates(self):
             sage: P
             (1 : 2 : 3 , 2 : 1 : 3)
         """
-        for i in range(self.codomain().ambient_space().num_components()):
+        for i in range(self.codomain().ambient_space().n_components()):
             self[i].normalize_coordinates()
 
     def dehomogenize(self, L):
@@ -352,7 +352,7 @@ def dehomogenize(self, L):
         PP = self.codomain()
         A = PP.affine_patch(L)
         pt = []
-        for i in range(PP.ambient_space().num_components()):
+        for i in range(PP.ambient_space().n_components()):
             pt.extend(self[i].dehomogenize(L[i]))
         return A(pt)
 
@@ -377,9 +377,9 @@ def scale_by(self, t):
         """
         if not isinstance(t, (tuple, list)):
             raise TypeError("%s must be a list or tuple" % t)
-        if len(t) != self.codomain().ambient_space().num_components():
+        if len(t) != self.codomain().ambient_space().n_components():
             raise TypeError("%s must have same number of components as %r" % (t, self))
-        for i in range(self.codomain().ambient_space().num_components()):
+        for i in range(self.codomain().ambient_space().n_components()):
             self[i].scale_by(t[i])
 
     def change_ring(self, R, **kwds):
@@ -463,7 +463,7 @@ def global_height(self, prec=None):
         if K not in NumberFields() and K != ZZ and not isinstance(K, (sage.rings.abc.Order, sage.rings.abc.AlgebraicField)):
             raise TypeError("must be over a number field or a number field order or QQbar")
 
-        n = self.codomain().ambient_space().num_components()
+        n = self.codomain().ambient_space().n_components()
         return max(self[i].global_height(prec=prec) for i in range(n))
 
     def local_height(self, v, prec=None):
@@ -501,7 +501,7 @@ def local_height(self, v, prec=None):
         if K not in NumberFields():
             raise TypeError("must be over a number field or a number field order")
 
-        n = self.codomain().ambient_space().num_components()
+        n = self.codomain().ambient_space().n_components()
         return max(self[i].local_height(v, prec=prec) for i in range(n))
 
 

src/sage/schemes/product_projective/rational_point.py

@@ -132,7 +132,7 @@ def enum_product_projective_rational_field(X, B):
             raise TypeError("codomain must be product of projective space over the rational field")
 
     R = X.codomain().ambient_space()
-    m = R.num_components()
+    m = R.n_components()
     iters = [R[i].points_of_bounded_height(bound=B) for i in range(m)]
     dim = [R[i].dimension_relative() + 1 for i in range(m)]
 
@@ -356,7 +356,7 @@ def sieve(X, bound):
     N = P.ngens()
     dim_scheme = X.dimension()
 
-    num_comp = P.num_components()
+    num_comp = P.n_components()
     comp_dim_relative = [P[i].dimension_relative() + 1 for i in range(num_comp)]
 
     dim_prefix = [0, comp_dim_relative[0]] # prefixes dim list