sage.schemes: Update # needs

sagemath · Sep 16, 2023 · 37d1671 · 37d1671
1 parent 485b707
commit 37d1671
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diff --git a/src/sage/schemes/toric/divisor.py b/src/sage/schemes/toric/divisor.py
@@ -1063,7 +1063,7 @@ def polyhedron(self):
             (A vertex at (0, 0),)
             sage: D.is_nef()
             False
-            sage: dP7.integrate( D.ch() * dP7.Td() )
+            sage: dP7.integrate(D.ch() * dP7.Td())                                      # needs sage.libs.singular
             1
             sage: P_antiK = (-dP7.K()).polyhedron(); P_antiK
             A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 5 vertices
@@ -1267,7 +1267,7 @@ def Kodaira_map(self, names='z'):
 
             sage: P1.<u,v> = toric_varieties.P1()
             sage: D = -P1.K()
-            sage: D.Kodaira_map()
+            sage: D.Kodaira_map()                                                       # needs fpylll
             Scheme morphism:
               From: 1-d CPR-Fano toric variety covered by 2 affine patches
               To:   Closed subscheme of Projective Space of dimension 2
@@ -1276,7 +1276,7 @@ def Kodaira_map(self, names='z'):
 
             sage: dP6 = toric_varieties.dP6()
             sage: D = -dP6.K()
-            sage: D.Kodaira_map(names='x')
+            sage: D.Kodaira_map(names='x')                                              # needs fpylll
             Scheme morphism:
               From: 2-d CPR-Fano toric variety covered by 6 affine patches
               To:   Closed subscheme of Projective Space of dimension 6

diff --git a/src/sage/schemes/toric/fano_variety.py b/src/sage/schemes/toric/fano_variety.py
@@ -1518,7 +1518,7 @@ def cohomology_class(self):
              covered by 8 affine patches defined by:
               a2*z0^2*z1 + a5*z0*z1*z3 + a1*z1*z3^2 + a3*z0^2*z4 + a4*z0*z3*z4 + a0*z3^2*z4,
               b1*z1*z2^2 + b2*z2^2*z4 + b5*z1*z2*z5 + b4*z2*z4*z5 + b3*z1*z5^2 + b0*z4*z5^2
-            sage: CI.cohomology_class()
+            sage: CI.cohomology_class()                                                 # needs sage.libs.singular
             [2*z3*z4 + 4*z3*z5 + 2*z4*z5]
         """
         X = self.ambient_space()

diff --git a/src/sage/schemes/toric/homset.py b/src/sage/schemes/toric/homset.py
@@ -621,6 +621,7 @@ def __iter__(self):
 
         EXAMPLES::
 
+            sage: # needs sage.libs.singular
             sage: P2.<x,y,z> = toric_varieties.P2(base_ring=GF(5))
             sage: cubic = P2.subscheme([x^3 + y^3 + z^3])
             sage: list(cubic.point_set())
@@ -641,6 +642,7 @@ def cardinality(self):
 
         EXAMPLES::
 
+            sage: # needs sage.libs.singular
             sage: P2.<x,y,z> = toric_varieties.P2(base_ring=GF(5))
             sage: cubic = P2.subscheme([x^3 + y^3 + z^3])
             sage: list(cubic.point_set())

diff --git a/src/sage/schemes/toric/points.py b/src/sage/schemes/toric/points.py
@@ -1007,16 +1007,16 @@ def cardinality(self):
             sage: X.<u,v,w> = ToricVariety(fan, base_ring=GF(7))
             sage: Y = X.subscheme(u^3 + v^3 + w^3 + u*v*w)
             sage: point_set = Y.point_set()
-            sage: list(point_set)
+            sage: list(point_set)                                                       # needs fpylll
             [[0 : 1 : 3],
              [1 : 0 : 3],
              [1 : 3 : 0],
              [1 : 1 : 6],
              [1 : 1 : 4],
              [1 : 3 : 2],
              [1 : 3 : 5]]
-            sage: ffe = point_set._enumerator()
-            sage: ffe.cardinality()
+            sage: ffe = point_set._enumerator()                                         # needs fpylll
+            sage: ffe.cardinality()                                                     # needs fpylll
             7
         """
         n = 0

diff --git a/src/sage/schemes/toric/toric_subscheme.py b/src/sage/schemes/toric/toric_subscheme.py
@@ -270,6 +270,7 @@ def affine_algebraic_patch(self, cone=None, names=None):
             Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
               z0^3 + z1^3 + 1
 
+            sage: # needs fpylll
             sage: cone = Cone([(0,1), (2,1)])
             sage: A2Z2.<x,y> = AffineToricVariety(cone)
             sage: A2Z2.affine_algebraic_patch()

diff --git a/src/sage/schemes/toric/variety.py b/src/sage/schemes/toric/variety.py
@@ -66,7 +66,7 @@
     sage: C2 = AffineToricVariety(quadrant, base_field=CC)
     sage: C2.base_ring()
     Complex Field with 53 bits of precision
-    sage: C2(1,2+i)
+    sage: C2(1, 2+i)                                                                    # needs sage.symbolic
     [1.00000000000000 : 2.00000000000000 + 1.00000000000000*I]
 
 or even ::
@@ -1083,7 +1083,7 @@ def coordinate_ring(self):
 
             sage: R = toric_varieties.A1().coordinate_ring();  R
             Multivariate Polynomial Ring in z over Rational Field
-            sage: type(R)
+            sage: type(R)                                                               # needs sage.libs.singular
             <... 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular'>
         """
         if "_coordinate_ring" not in self.__dict__:
@@ -1931,6 +1931,7 @@ def cohomology_ring(self):
         potentially troublesome on unpickling, see :trac:`15050`
         and :trac:`15149` ::
 
+            sage: # needs sage.libs.singular
             sage: variety = toric_varieties.P(1)
             sage: a = [variety.cohomology_ring(), variety.cohomology_basis(), variety.volume_class()]
             sage: b = [variety.Todd_class(), variety.Chern_class(), variety.Chern_character(), variety.Kaehler_cone(), variety.Mori_cone()]
@@ -2077,6 +2078,7 @@ def integrate(self, cohomology_class):
 
         EXAMPLES::
 
+            sage: # needs sage.libs.singular
             sage: dP6 = toric_varieties.dP6()
             sage: HH = dP6.cohomology_ring()
             sage: D = [ HH(c) for c in dP6.fan(dim=1) ]
@@ -2098,6 +2100,7 @@ def integrate(self, cohomology_class):
         If the toric variety is an orbifold, the intersection numbers
         are usually fractional::
 
+            sage: # needs sage.libs.singular
             sage: P2_123 = toric_varieties.P2_123()
             sage: HH = P2_123.cohomology_ring()
             sage: D = [ HH(c) for c in P2_123.fan(dim=1) ]
@@ -2159,6 +2162,7 @@ def Chern_class(self, deg=None):
 
         EXAMPLES::
 
+            sage: # needs sage.libs.singular
             sage: X = toric_varieties.dP6()
             sage: X.Chern_class()
             [-6*w^2 + y + 2*v + 2*z + w + 1]
@@ -2204,6 +2208,7 @@ def Chern_character(self, deg=None):
 
         EXAMPLES::
 
+            sage: # needs sage.libs.singular
             sage: dP6 = toric_varieties.dP6()
             sage: dP6.Chern_character()
             [3*w^2 + y + 2*v + 2*z + w + 2]
@@ -2243,6 +2248,7 @@ def Todd_class(self, deg=None):
 
         EXAMPLES::
 
+            sage: # needs sage.libs.singular
             sage: dP6 = toric_varieties.dP6()
             sage: dP6.Todd_class()
             [-w^2 + 1/2*y + v + z + 1/2*w + 1]
@@ -2289,6 +2295,7 @@ def Euler_number(self):
 
         EXAMPLES::
 
+            sage: # needs sage.libs.singular
             sage: P1xP1 = toric_varieties.P1xP1()
             sage: P1xP1.Euler_number()
             4
@@ -2320,7 +2327,7 @@ def K(self):
             sage: HH = dP6.cohomology_ring()
             sage: dP6.K()
             -V(x) - V(u) - V(y) - V(v) - V(z) - V(w)
-            sage: dP6.integrate( HH(dP6.K())^2 )
+            sage: dP6.integrate( HH(dP6.K())^2 )                                        # needs sage.libs.singular
             6
         """
         from sage.schemes.toric.divisor import ToricDivisor
@@ -2479,18 +2486,18 @@ def _semigroup_ring(self, cone=None, names=None):
         EXAMPLES::
 
             sage: A2Z2 = Cone([(0,1), (2,1)])
-            sage: AffineToricVariety(A2Z2)._semigroup_ring()
+            sage: AffineToricVariety(A2Z2)._semigroup_ring()                            # needs fpylll
             (Multivariate Polynomial Ring in z0, z1, z2 over Rational Field,
              Ideal (-z0*z1 + z2^2) of Multivariate Polynomial Ring in z0, z1, z2 over Rational Field,
              2-d cone in 2-d lattice M)
 
              sage: P2 = toric_varieties.P2()
              sage: cone = P2.fan().generating_cone(0)
-             sage: P2._semigroup_ring(cone)
+             sage: P2._semigroup_ring(cone)                                             # needs fpylll
              (Multivariate Polynomial Ring in z0, z1 over Rational Field,
               Ideal (0) of Multivariate Polynomial Ring in z0, z1 over Rational Field,
               2-d cone in 2-d lattice M)
-             sage: P2.change_ring(GF(101))._semigroup_ring(cone)
+             sage: P2.change_ring(GF(101))._semigroup_ring(cone)                        # needs fpylll
              (Multivariate Polynomial Ring in z0, z1 over Finite Field of size 101,
               Ideal (0) of Multivariate Polynomial Ring in z0, z1 over Finite Field of size 101,
               2-d cone in 2-d lattice M)
@@ -2560,7 +2567,7 @@ def Spec(self, cone=None, names=None):
         A more interesting example::
 
             sage: A2Z2 = Cone([(0,1), (2,1)])
-            sage: AffineToricVariety(A2Z2).Spec(names='u,v,t')
+            sage: AffineToricVariety(A2Z2).Spec(names='u,v,t')                          # needs fpylll
             Spectrum of Quotient of Multivariate Polynomial Ring
             in u, v, t over Rational Field by the ideal (-u*v + t^2)
         """
@@ -2594,10 +2601,10 @@ def affine_algebraic_patch(self, cone=None, names=None):
 
             sage: cone = Cone([(0,1), (2,1)])
             sage: A2Z2 = AffineToricVariety(cone)
-            sage: A2Z2.affine_algebraic_patch()
+            sage: A2Z2.affine_algebraic_patch()                                         # needs fpylll
             Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
               -z0*z1 + z2^2
-            sage: A2Z2.affine_algebraic_patch(Cone([(0,1)]), names='x, y, t')
+            sage: A2Z2.affine_algebraic_patch(Cone([(0,1)]), names='x, y, t')           # needs fpylll
             Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
               1
         """
@@ -3085,7 +3092,7 @@ def __init__(self, variety):
             True
             sage: TDiv.scheme().cohomology_ring() is X2.cohomology_ring()  # this is where it gets tricky
             True
-            sage: TDiv.gen(0).Chern_character() * X2.cohomology_ring().one()
+            sage: TDiv.gen(0).Chern_character() * X2.cohomology_ring().one()            # needs sage.libs.singular
             [1]
         """
         self._variety = variety
@@ -3147,6 +3154,7 @@ def _element_constructor_(self,x):
 
         EXAMPLES::
 
+            sage: # needs sage.libs.singular
             sage: dP6 = toric_varieties.dP6()
             sage: H = dP6.cohomology_ring()
             sage: cone = dP6.fan().cone_containing(2,3); cone
@@ -3164,6 +3172,7 @@ def _element_constructor_(self,x):
         coefficient is a multiple depending on the orbifold
         singularity. See also [CLS2011]_, Lemma 12.5.2::
 
+            sage: # needs sage.libs.singular
             sage: P2_123 = toric_varieties.P2_123()
             sage: HH = P2_123.cohomology_ring()
             sage: HH(Cone([(1,0)])) * HH(Cone([(-2,-3)]))
@@ -3179,6 +3188,7 @@ def _element_constructor_(self,x):
 
         Numbers will be converted into the ring::
 
+            sage: # needs sage.libs.singular
             sage: P2 = toric_varieties.P2()
             sage: H = P2.cohomology_ring()
             sage: H._element_constructor_(1)
@@ -3222,6 +3232,7 @@ def __call__(self, x, coerce=True):
 
         EXAMPLES::
 
+            sage: # needs sage.libs.singular
             sage: P2 = toric_varieties.P2()
             sage: H = P2.cohomology_ring()
             sage: H(1)
@@ -3245,7 +3256,7 @@ def gens(self):
         EXAMPLES::
 
             sage: P2 = toric_varieties.P2()
-            sage: P2.cohomology_ring().gens()
+            sage: P2.cohomology_ring().gens()                                           # needs sage.libs.singular
             ([z], [z], [z])
         """
         if "_gens" not in self.__dict__:
@@ -3270,7 +3281,7 @@ def gen(self, i):
         EXAMPLES::
 
             sage: P2 = toric_varieties.P2()
-            sage: P2.cohomology_ring().gen(2)
+            sage: P2.cohomology_ring().gen(2)                                           # needs sage.libs.singular
             [z]
         """
         return CohomologyClass(self, self._polynomial_ring.gen(i))
@@ -3295,9 +3306,9 @@ def is_CohomologyClass(x):
         sage: P2 = toric_varieties.P2()
         sage: HH = P2.cohomology_ring()
         sage: from sage.schemes.toric.variety import is_CohomologyClass
-        sage: is_CohomologyClass( HH.one() )
+        sage: is_CohomologyClass( HH.one() )                                            # needs sage.libs.singular
         True
-        sage: is_CohomologyClass( HH(P2.fan(1)[0]) )
+        sage: is_CohomologyClass( HH(P2.fan(1)[0]) )                                    # needs sage.libs.singular
         True
         sage: is_CohomologyClass('z')
         False
@@ -3319,6 +3330,7 @@ class CohomologyClass(QuotientRingElement):
 
     EXAMPLES::
 
+        sage: # needs sage.libs.singular
         sage: P2 = toric_varieties.P2()
         sage: P2.cohomology_ring().gen(0)
         [z]
@@ -3348,7 +3360,7 @@ def __init__(self, cohomology_ring, representative):
             sage: P2 = toric_varieties.P2()
             sage: H = P2.cohomology_ring()
             sage: from sage.schemes.toric.variety import CohomologyClass
-            sage: CohomologyClass(H, H.defining_ideal().ring().zero() )
+            sage: CohomologyClass(H, H.defining_ideal().ring().zero() )                 # needs sage.libs.singular
             [0]
         """
         assert representative in cohomology_ring.defining_ideal().ring(), \
@@ -3365,7 +3377,7 @@ def _repr_(self):
 
         EXAMPLES::
 
-            sage: toric_varieties.P2().cohomology_ring().gen(0)._repr_()
+            sage: toric_varieties.P2().cohomology_ring().gen(0)._repr_()                # needs sage.libs.singular
             '[z]'
         """
         return '[' + super()._repr_() + ']'
@@ -3380,8 +3392,8 @@ def _latex_(self):
 
         EXAMPLES::
 
-            sage: cohomology_class = toric_varieties.P2().cohomology_ring().gen(0)^2/2
-            sage: cohomology_class._latex_()
+            sage: cohomology_class = toric_varieties.P2().cohomology_ring().gen(0)^2/2  # needs sage.libs.singular
+            sage: cohomology_class._latex_()                                            # needs sage.libs.singular
             '\\left[ \\frac{1}{2} z^{2} \\right]'
         """
         return r'\left[ %s \right]' % latex(self.lift())
@@ -3399,9 +3411,9 @@ def deg(self):
         EXAMPLES::
 
             sage: P2 = toric_varieties.P2()
-            sage: P2.cohomology_ring().gen(0).deg()
+            sage: P2.cohomology_ring().gen(0).deg()                                     # needs sage.libs.singular
             1
-            sage: P2.cohomology_ring().zero().deg()
+            sage: P2.cohomology_ring().zero().deg()                                     # needs sage.libs.singular
             -1
         """
         return self.lift().degree()
@@ -3424,6 +3436,7 @@ def part_of_degree(self, d):
 
         EXAMPLES::
 
+            sage: # needs sage.libs.singular
             sage: P1xP1 = toric_varieties.P1xP1()
             sage: t = P1xP1.cohomology_ring().gen(0)
             sage: y = P1xP1.cohomology_ring().gen(2)
@@ -3460,6 +3473,7 @@ def exp(self):
 
         EXAMPLES::
 
+            sage: # needs sage.libs.singular
             sage: P2 = toric_varieties.P2()
             sage: H_class = P2.cohomology_ring().gen(0)
             sage: H_class

diff --git a/src/sage/schemes/toric/weierstrass_covering.py b/src/sage/schemes/toric/weierstrass_covering.py
@@ -351,7 +351,7 @@ def WeierstrassMap_P1xP1(polynomial, variables=None):
         sage: f, g = WeierstrassForm_P1xP1(biquadric, [x0, x1, y0, y1]);  (f,g)
         (-625/48*a^4 + 25/3*a^2 - 16/3, 15625/864*a^6 - 625/36*a^4 - 100/9*a^2 + 128/27)
         sage: X, Y, Z = WeierstrassMap_P1xP1(biquadric, [x0, x1, y0, y1])
-        sage: (-Y^2 + X^3 + f*X*Z^4 + g*Z^6).reduce(R.ideal(biquadric))
+        sage: (-Y^2 + X^3 + f*X*Z^4 + g*Z^6).reduce(R.ideal(biquadric))                 # needs sage.libs.singular
         0
 
         sage: R = PolynomialRing(QQ, 'x,y,s,t', order='lex')
@@ -361,7 +361,7 @@ def WeierstrassMap_P1xP1(polynomial, variables=None):
         ....:             + t^2*(7*x^2+8*x*y+9*y^2))
         sage: X, Y, Z = WeierstrassMap_P1xP1(equation, [x,y,s,t])
         sage: f, g = WeierstrassForm_P1xP1(equation, variables=[x,y,s,t])
-        sage: (-Y^2 + X^3 + f*X*Z^4 + g*Z^6).reduce(R.ideal(equation))
+        sage: (-Y^2 + X^3 + f*X*Z^4 + g*Z^6).reduce(R.ideal(equation))                  # needs sage.libs.singular
         0
 
         sage: R = PolynomialRing(QQ, 'x,s', order='lex')
@@ -370,7 +370,7 @@ def WeierstrassMap_P1xP1(polynomial, variables=None):
         sage: equation = s^2*(x^2+2*x+3) + s*(4*x^2+5*x+6) + (7*x^2+8*x+9)
         sage: X, Y, Z = WeierstrassMap_P1xP1(equation)
         sage: f, g = WeierstrassForm_P1xP1(equation)
-        sage: (-Y^2 + X^3 + f*X*Z^4 + g*Z^6).reduce(R.ideal(equation))
+        sage: (-Y^2 + X^3 + f*X*Z^4 + g*Z^6).reduce(R.ideal(equation))                  # needs sage.libs.singular
         0
     """
     x, y, s, t = _check_polynomial_P1xP1(polynomial, variables)
@@ -425,7 +425,7 @@ def WeierstrassMap_P2_112(polynomial, variables=None):
         sage: equation = y^2 + a0*x^4 + 4*a1*x^3 + 6*a2*x^2 + 4*a3*x + a4
         sage: X, Y, Z = WeierstrassMap_P2_112(equation, [x,y])
         sage: f, g = WeierstrassForm_P2_112(equation, variables=[x,y])
-        sage: (-Y^2 + X^3 + f*X*Z^4 + g*Z^6).reduce(R.ideal(equation))
+        sage: (-Y^2 + X^3 + f*X*Z^4 + g*Z^6).reduce(R.ideal(equation))                  # needs sage.libs.singular
         0
 
     Another example, this time in homogeneous coordinates::
@@ -441,7 +441,7 @@ def WeierstrassMap_P2_112(polynomial, variables=None):
         sage: WeierstrassForm_P2_112(C_eqn, [x,y,z,t])
         (-97/48, 17/864)
         sage: X, Y, Z = WeierstrassMap_P2_112(C_eqn, [x,y,z,t])
-        sage: (-Y^2 + X^3 - 97/48*X*Z^4 + 17/864*Z^6).reduce(C.defining_ideal())
+        sage: (-Y^2 + X^3 - 97/48*X*Z^4 + 17/864*Z^6).reduce(C.defining_ideal())        # needs sage.libs.singular
         0
     """
     x, y, z, t = _check_polynomial_P2_112(polynomial, variables)