Improve genus method of function fields and curves

src/sage/rings/function_field/function_field_polymod.py

@@ -34,6 +34,7 @@
 from sage.rings.integer import Integer
 from sage.categories.homset import Hom
 from sage.categories.function_fields import FunctionFields
+from sage.categories.number_fields import NumberFields
 
 from .element import FunctionFieldElement
 from .element_polymod import FunctionFieldElement_polymod
@@ -1842,11 +1843,27 @@ def genus(self):
             sage: L.genus()
             6
 
+            sage: # needs sage.rings.number_field
+            sage: R.<T> = QQ[]
+            sage: N.<a> = NumberField(T^2 + 1)
+            sage: K.<x> = FunctionField(N); K
+            Rational function field in x over Number Field in a with defining polynomial T^2 + 1
+            sage: K.genus()
+            0
+            sage: S.<t> = PolynomialRing(K)
+            sage: L.<y> = K.extension(t^2 - x^3 + x)
+            sage: L.genus()
+            1
+
         The genus is computed by the Hurwitz genus formula.
         """
         k, _ = self.exact_constant_field()
+        if k in NumberFields():
+            k_degree = k.relative_degree()
+        else:
+            k_degree = k.degree()
         different_degree = self.different().degree()  # must be even
-        return Integer(different_degree // 2 - self.degree() / k.degree()) + 1
+        return Integer(different_degree // 2 - self.degree() / k_degree) + 1
 
     def residue_field(self, place, name=None):
         """

src/sage/rings/polynomial/multi_polynomial_ideal.py

@@ -1569,13 +1569,20 @@ def _groebner_basis_singular_raw(self, algorithm="groebner", singular=singular_d
     @handle_AA_and_QQbar
     def genus(self):
         r"""
-        Return the genus of the projective curve defined by this ideal,
-        which must be 1 dimensional.
+        Return the geometric genus of the projective curve defined by this
+        ideal.
+
+        OUTPUT:
+
+        If the ideal is homogeneous and defines a curve in a projective space,
+        then the genus of the curve is returned. If the ideal is not
+        homogeneous and defines a curve in an affine space, the genus of the
+        projective closure of the curve is returned.
 
         EXAMPLES:
 
-        Consider the hyperelliptic curve `y^2 = 4x^5 - 30x^3 + 45x -
-        22` over `\QQ`, it has genus 2::
+        Consider the hyperelliptic curve `y^2 = 4x^5 - 30x^3 + 45x - 22` over
+        `\QQ`, it has genus 2::
 
             sage: P.<x> = QQ[]
             sage: f = 4*x^5 - 30*x^3 + 45*x - 22
@@ -1592,28 +1599,36 @@ def genus(self):
             sage: I.genus()
             2
 
-        TESTS:
-
-        Check that the answer is correct for reducible curves::
+        Geometric genus is only defined for geometrically irreducible curves.
+        You may get a nonsensical answer if the condition is not met.  A curve
+        reducible over a quadratic extension of `\QQ`::
 
             sage: R.<x, y, z> = QQ[]
             sage: C = Curve(x^2 - 2*y^2)
-            sage: C.is_singular()
-            True
             sage: C.genus()
             -1
-            sage: Ideal(x^4+y^2*x+x).genus()
+
+        TESTS:
+
+        An ideal that does not define a curve but we get a result! ::
+
+            sage: R.<x, y, z> = QQ[]
+            sage: Ideal(x^4 + y^2*x + x).genus()
             0
+
+        An ideal that defines a geometrically reducible affine curve::
+
             sage: T.<t1,t2,u1,u2> = QQ[]
             sage: TJ = Ideal([t1^2 + u1^2 - 1,t2^2 + u2^2 - 1, (t1-t2)^2 + (u1-u2)^2 -1])
             sage: TJ.genus()
             -1
 
         Check that this method works over QQbar (:trac:`25351`)::
 
-            sage: P.<x,y> = QQbar[]                                                     # needs sage.rings.number_field
-            sage: I = ideal(y^3*z + x^3*y + x*z^3)                                      # needs sage.rings.number_field
-            sage: I.genus()                                                             # needs sage.rings.number_field
+            sage: # needs sage.rings.number_field
+            sage: P.<x,y> = QQbar[]
+            sage: I = ideal(y^3*z + x^3*y + x*z^3)
+            sage: I.genus()
             3
         """
         from sage.libs.singular.function_factory import ff

src/sage/schemes/curves/affine_curve.py

@@ -2065,13 +2065,21 @@ def _genus(self):
             sage: C = Curve(x^5 + y^5 + x*y + 1)
             sage: C.genus()   # indirect doctest
             1
-        """
-        k = self.base_ring()
 
+        TESTS::
+
+            sage: # needs sage.rings.number_field
+            sage: R.<T> = QQ[]
+            sage: N.<a> = NumberField(T^2 + 1)
+            sage: A2.<x,y> = AffineSpace(N, 2)
+            sage: C = Curve(y^2 - x^3 + x, A2)
+            sage: C.genus()
+            1
+        """
         # Singular's genus command is usually much faster than the genus method
         # of function fields in Sage. But unfortunately Singular's genus
-        # command does not yet work over non-prime finite fields.
-        if k.is_finite() and k.degree() > 1:
+        # command does not work over extension fields.
+        if self.base_ring().degree() > 1:
             return self._function_field.genus()
 
         # call Singular's genus command

src/sage/schemes/curves/constructor.py

@@ -37,24 +37,20 @@
 # ********************************************************************
 
 from sage.categories.fields import Fields
+from sage.categories.number_fields import NumberFields
 
 from sage.rings.polynomial.multi_polynomial import MPolynomial
 from sage.rings.polynomial.multi_polynomial_ring import is_MPolynomialRing
 from sage.rings.finite_rings.finite_field_base import FiniteField
-
 from sage.rings.rational_field import QQ
 
 from sage.structure.all import Sequence
 
-from sage.schemes.affine.affine_space import is_AffineSpace
 from sage.schemes.generic.ambient_space import is_AmbientSpace
 from sage.schemes.generic.algebraic_scheme import is_AlgebraicScheme
-from sage.schemes.projective.projective_space import is_ProjectiveSpace
-
-from sage.schemes.affine.affine_space import AffineSpace
-
-from sage.schemes.projective.projective_space import ProjectiveSpace
-
+from sage.schemes.affine.affine_space import AffineSpace, is_AffineSpace
+from sage.schemes.projective.projective_space import ProjectiveSpace, is_ProjectiveSpace
+from sage.schemes.plane_conics.constructor import Conic
 
 from .projective_curve import (ProjectiveCurve,
                                ProjectivePlaneCurve,
@@ -77,9 +73,6 @@
                            IntegralAffinePlaneCurve_finite_field)
 
 
-from sage.schemes.plane_conics.constructor import Conic
-
-
 def _is_irreducible_and_reduced(F) -> bool:
     """
     Check if the polynomial F is irreducible and reduced.
@@ -113,11 +106,13 @@
 
     INPUT:
 
-    - ``F`` -- a multivariate polynomial, or a list or tuple of polynomials, or an algebraic scheme.
+    - ``F`` -- a multivariate polynomial, or a list or tuple of polynomials, or an algebraic scheme
+
+    - ``A`` -- (default: None) an ambient space in which to create the curve
 
-    - ``A`` -- (default: None) an ambient space in which to create the curve.
+    EXAMPLES:
 
-    EXAMPLES: A projective plane curve.  ::
+    A projective plane curve::
 
         sage: x,y,z = QQ['x,y,z'].gens()
         sage: C = Curve(x^3 + y^3 + z^3); C
@@ -215,7 +210,7 @@
         sage: Curve(P1)
         Projective Line over Finite Field of size 5
 
-    ::
+    An affine line::
 
         sage: A1.<x> = AffineSpace(1, QQ)
         sage: R = A1.coordinate_ring()
@@ -224,6 +219,18 @@
         sage: Curve(A1)
         Affine Line over Rational Field
 
+    A projective line::
+
+        sage: R.<x> = QQ[]
+        sage: N.<a> = NumberField(x^2 + 1)
+        sage: P1.<x,y> = ProjectiveSpace(N, 1)
+        sage: C = Curve(P1)
+        sage: C
+        Projective Line over Number Field in a with defining polynomial x^2 + 1
+        sage: C.geometric_genus()
+        0
+        sage: C.arithmetic_genus()
+        0
     """
     if A is None:
         if is_AmbientSpace(F) and F.dimension() == 1:
@@ -298,12 +305,20 @@
     k = A.base_ring()
 
     if is_AffineSpace(A):
+        if n == 1:
+            if A.coordinate_ring().ideal(F).is_zero():
+                if isinstance(k, FiniteField):
+                    return IntegralAffineCurve_finite_field(A, F)
+                if k in Fields():
+                    return IntegralAffineCurve(A, F)
+                return AffineCurve(A, F)
+            raise TypeError(f"{F} does not define a curve in one-dimensional affine space")
         if n != 2:
             if isinstance(k, FiniteField):
                 if A.coordinate_ring().ideal(F).is_prime():
                     return IntegralAffineCurve_finite_field(A, F)
             if k in Fields():
-                if k == QQ and A.coordinate_ring().ideal(F).is_prime():
+                if (k == QQ or k in NumberFields()) and A.coordinate_ring().ideal(F).is_prime():
                     return IntegralAffineCurve(A, F)
                 return AffineCurve_field(A, F)
             return AffineCurve(A, F)
@@ -317,20 +332,28 @@
                 return IntegralAffinePlaneCurve_finite_field(A, F)
             return AffinePlaneCurve_finite_field(A, F)
         if k in Fields():
-            if k == QQ and _is_irreducible_and_reduced(F):
+            if (k == QQ or k in NumberFields()) and _is_irreducible_and_reduced(F):
                 return IntegralAffinePlaneCurve(A, F)
             return AffinePlaneCurve_field(A, F)
         return AffinePlaneCurve(A, F)
 
     elif is_ProjectiveSpace(A):
+        if n == 1:
+            if A.coordinate_ring().ideal(F).is_zero():
+                if isinstance(k, FiniteField):
+                    return IntegralProjectiveCurve_finite_field(A, F)
+                if k in Fields():
+                    return IntegralProjectiveCurve(A, F)
+                return ProjectiveCurve(A, F)
+            raise TypeError(f"{F} does not define a curve in one-dimensional projective space")
         if n != 2:
             if not all(f.is_homogeneous() for f in F):
                 raise TypeError("polynomials defining a curve in a projective space must be homogeneous")
             if isinstance(k, FiniteField):
                 if A.coordinate_ring().ideal(F).is_prime():
                     return IntegralProjectiveCurve_finite_field(A, F)
             if k in Fields():
-                if k == QQ and A.coordinate_ring().ideal(F).is_prime():
+                if (k == QQ or k in NumberFields()) and A.coordinate_ring().ideal(F).is_prime():
                     return IntegralProjectiveCurve(A, F)
                 return ProjectiveCurve_field(A, F)
             return ProjectiveCurve(A, F)
@@ -349,7 +372,7 @@
                 return IntegralProjectivePlaneCurve_finite_field(A, F)
             return ProjectivePlaneCurve_finite_field(A, F)
         if k in Fields():
-            if k == QQ and _is_irreducible_and_reduced(F):
+            if (k == QQ or k in NumberFields()) and _is_irreducible_and_reduced(F):
                 return IntegralProjectivePlaneCurve(A, F)
             return ProjectivePlaneCurve_field(A, F)
         return ProjectivePlaneCurve(A, F)

src/sage/schemes/curves/curve.py

@@ -209,17 +209,9 @@
         r"""
         Return the geometric genus of the curve.
 
-        This is by definition the genus of the normalization of the projective
-        closure of the curve over the algebraic closure of the base field; the
-        base field must be a prime field.
-
-        .. NOTE::
-
-            This calls Singular's genus command.
-
         EXAMPLES:
 
-        Examples of projective curves. ::
+        Examples of projective curves::
 
             sage: P2 = ProjectiveSpace(2, GF(5), names=['x','y','z'])
             sage: x, y, z = P2.coordinate_ring().gens()
@@ -233,7 +225,7 @@
             sage: C.geometric_genus()
             3
 
-        Examples of affine curves. ::
+        Examples of affine curves::
 
             sage: x, y = PolynomialRing(GF(5), 2, 'xy').gens()
             sage: C = Curve(y^2 - x^3 - 17*x + y)
@@ -246,12 +238,22 @@
             sage: C.geometric_genus()
             3
 
+        .. WARNING::
+
+            Geometric genus is only defined for `geometrically irreducible curve
+            <https://stacks.math.columbia.edu/tag/0BYE>`_. This method does not
+            check the condition. You may get a nonsensical result if the curve is
+            not geometrically irreducible::
+
+                sage: P2.<x,y,z> = ProjectiveSpace(QQ, 2)
+                sage: C = Curve(x^2 + y^2, P2)
+                sage: C.geometric_genus()  # nonsense!
+                -1
         """
         try:
             return self._genus
         except AttributeError:
-            self._genus = self.defining_ideal().genus()
-            return self._genus
+            raise NotImplementedError
 
     def union(self, other):
         """