diff --git a/src/sage/schemes/curves/affine_curve.py b/src/sage/schemes/curves/affine_curve.py
index 8d6474115ef..5eefb3f8e99 100644
--- a/src/sage/schemes/curves/affine_curve.py
+++ b/src/sage/schemes/curves/affine_curve.py
@@ -2143,7 +2143,7 @@ def function(self, f):
         if f not in R and f.parent() is self.coordinate_ring():
             f = f.lift()
 
-        phi = self._lift_to_function_field
+        phi = self._map_to_function_field
         num = R(f.numerator())
         den = R(f.denominator())
         return phi(num) / phi(den)
@@ -2373,7 +2373,7 @@ def _function_field(self):
         return self._nonsingular_model[0]
 
     @lazy_attribute
-    def _lift_to_function_field(self):
+    def _map_to_function_field(self):
         """
         Return the map to the function field of the curve.
 
@@ -2381,7 +2381,7 @@ def _lift_to_function_field(self):
 
             sage: A.<x,y,z> = AffineSpace(GF(11), 3)
             sage: C = Curve([x*z - y^2, y - z^2, x - y*z], A)
-            sage: C._lift_to_function_field
+            sage: C._map_to_function_field
             Ring morphism:
               From: Fraction Field of Multivariate Polynomial Ring in x, y, z
                     over Finite Field of size 11
@@ -2438,7 +2438,7 @@ def _singularities(self):
             [(Point (x, y),
               [Place (x, 1/x*y^3 + 1/x*y^2 + 1), Place (x, 1/x*y^3 + 1/x*y^2 + 6)])]
         """
-        to_F = self._lift_to_function_field
+        to_F = self._map_to_function_field
         sing = self.singular_subscheme()
 
         funcs = []
@@ -2655,7 +2655,7 @@ def places_on(self, point):
             sage: Cp = Curve(x^3*y + y^3*z + x*z^3)
             sage: C = Cp.affine_patch(0)
         """
-        phi = self._lift_to_function_field
+        phi = self._map_to_function_field
         gs = [phi(g) for g in point.prime_ideal().gens()]
         fs = [g for g in gs if not g.is_zero()]
         f = fs.pop()
diff --git a/src/sage/schemes/curves/projective_curve.py b/src/sage/schemes/curves/projective_curve.py
index 3e6c40920c1..21efc6b6109 100644
--- a/src/sage/schemes/curves/projective_curve.py
+++ b/src/sage/schemes/curves/projective_curve.py
@@ -2349,7 +2349,7 @@ def function(self, f):
              - Place (y, z + 1)
         """
         S = self.ambient_space().coordinate_ring()
-        phi = self._lift_to_function_field
+        phi = self._map_to_function_field
         num = S(f.numerator())
         den = S(f.denominator())
         if num.degree() != den.degree():
@@ -2427,7 +2427,7 @@ def _function_field(self):
         return self._open_affine._function_field
 
     @lazy_attribute
-    def _lift_to_function_field(self):
+    def _map_to_function_field(self):
         """
         Return the map to function field of the curve.
 
@@ -2435,7 +2435,7 @@ def _lift_to_function_field(self):
 
             sage: P.<x,y,z> = ProjectiveSpace(GF(5), 2)
             sage: C = Curve(y^2*z^7 - x^9 - x*z^8)
-            sage: C._lift_to_function_field
+            sage: C._map_to_function_field
             Ring morphism:
               From: Multivariate Polynomial Ring in x, y, z over Finite Field of size 5
               To:   Function field in z defined by z^8 + 4*y^2*z^7 + 1
@@ -2508,7 +2508,7 @@ def _singularities(self):
 
         """
         S = self.ambient_space().coordinate_ring()
-        to_F = self._lift_to_function_field
+        to_F = self._map_to_function_field
         sing = self.singular_subscheme()  # singular locus
 
         # for each affine patch, places on which the dehomogenized polynomials
@@ -2694,7 +2694,7 @@ def places_on(self, point):
             if not S.gen(i) in prime:
                 break
 
-        phi = self._lift_to_function_field
+        phi = self._map_to_function_field
         denom = self._coordinate_functions[i]
         gs = [phi(f)/denom**f.degree() for f in prime.gens()]
         fs = [g for g in gs if not g.is_zero()]