Ticket #8988: trac_8988_Kahlercone_lattice.patch

sage/schemes/generic/toric_variety.py

@@ -231 +231 @@
 from sage.geometry.all import Cone, Fan
 from sage.matrix.all import matrix
 from sage.misc.all import latex
+from sage.modules.free_module import FreeModule
 from sage.modules.free_module_element import vector
-from sage.rings.all import PolynomialRing, QQ, is_Field, is_FractionField
+from sage.rings.all import PolynomialRing, ZZ, QQ, is_Field, is_FractionField
 from sage.schemes.generic.algebraic_scheme import AlgebraicScheme_subscheme
 from sage.schemes.generic.ambient_space import AmbientSpace
 from sage.schemes.generic.homset import SchemeHomset_coordinates
@@ -1183 +1184 @@
         """
         return self.fan().is_smooth()
 
-    def kaehler_cone(self):
+    def rational_divisor_class_group(self):
         r"""
-        Return the Kaehler cone of ``self``.
+        The rational divisor class group.
 
         OUTPUT:
 
-        - matrix.
+        - Returns the divisor class group modulo torsion,
+          `\mathop{Cl}(X) \otimes_\ZZ \QQ`, of the toric variety.
+
+        .. NOTE::
+        
+            * If the toric variety is **smooth**, this equals the
+              Picard group: the elements are the isomorphism classes
+              of line bundles on the toric variety. The group law
+              (which we write as addition) is the tensor product of
+              the line bundles. The Picard group of a toric variety is
+              always torsion-free.
+        
+            * The Weil divisor class group `\mathop{Cl}(X)` of a toric
+              variety is a finitely generated abelian group and can
+              contain torsion. Its rank equals (number of rays)-(rank
+              of M,N-lattice).
+
+            * The coordinates correspond to the rows of ``self.fan().gale_transform()``
+        
+            * :meth:`Kaehler_cone` yields a cone in this group.
+
+        EXAMPLES::
+        
+            sage: fan = FaceFan(lattice_polytope.octahedron(2))
+            sage: P1xP1 = ToricVariety(fan)
+            sage: P1xP1.rational_divisor_class_group()
+            Vector space of dimension 2 over Rational Field
+        """
+        if "_rational_divisor_class_group" not in self.__dict__:
+            self._rational_divisor_class_group = \
+                FreeModule(QQ, self.fan().nrays()-self.fan().lattice_dim())
+        return self._rational_divisor_class_group
+
+    def Kaehler_cone(self):
+        r"""
+        The Kähler cone
+
+        OUTPUT:
+
+        Returns the Kähler cone of ``self`` as a cone in
+        `\mathop{Cl}(X) \otimes_\ZZ \QQ`, see
+        :meth:`rational_divisor_class_group`.
 
         EXAMPLES::
 
             sage: fan = FaceFan(lattice_polytope.octahedron(2))
             sage: P1xP1 = ToricVariety(fan)
-            sage: P1xP1.kaehler_cone().ray_matrix()
-            [0 1]
-            [1 0]
+            sage: Kc = P1xP1.Kaehler_cone()
+            sage: Kc
+            2-d cone in 2-d lattice
+            sage: Kc.rays()
+            ((0, 1), (1, 0))
+            sage: Kc.lattice()
+            Vector space of dimension 2 over Rational Field
         """
-        if "_kaehler_cone" not in self.__dict__:
+        if "_Kaehler_cone" not in self.__dict__:
             fan = self.fan()
             GT = fan.gale_transform().columns()
             n = fan.nrays()
             K = None
             for cone in fan:
-                sigma = Cone(GT[i] for i in range(n)
-                                   if i not in cone.ambient_ray_indices())
+                sigma = Cone([GT[i] for i in range(n)
+                                    if i not in cone.ambient_ray_indices()],
+                             lattice = self.rational_divisor_class_group())
                 K = K.intersection(sigma) if K is not None else sigma
-            self._kaehler_cone = K
-        return self._kaehler_cone
+            self._Kaehler_cone = K
+        return self._Kaehler_cone
+
+    def Mori_vectors(self):
+        """
+        Returns the rays of the Mori cone.
+
+        OUTPUT:
+
+        - The rays of the Mori cone, that is, the dual of the Kähler cone. 
+
+        - The points in the Mori cone are the effective curves in the variety.
+
+        - The first ``self.fan().nrays()`` integer entries in each
+          Mori vector are the intersection numbers of the curve
+          corresponding to the generator of the ray with the divisors
+          of the toric variety in the same order as :meth:`divisors`.
+
+        - The last entry is associated to the orgin of the N-lattice.
+
+        - The Mori vectors are also known as the gauged linear sigma
+          model charge vectors.
+
+        EXAMPLES::
+
+            sage: P4_11169 = toric_varieties.P4_11169_resolved()
+            sage: P4_11169.Mori_vectors()
+            [(3, 2, 0, 0, 0, 1, -6), (0, 0, 1, 1, 1, -3, 0)]
+        """
+        Mc = [ facet_normal * self._fan.gale_transform() 
+               for facet_normal in self.Kaehler_cone().facet_normals() ]
+        return Mc;
 
     def resolve(self, **kwds):
         r"""