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NormalToricVarieties :: fromPicToCl

fromPicToCl -- get the map from Picard group to class group

Synopsis

Description

The Picard group of a normal toric variety is a subgroup of the class group.

On a smooth normal toric variety, the Picard group is isomorphic to the class group, so the inclusion map is the identity.

i1 : PP3 = projectiveSpace 3;
i2 : pic PP3

       1
o2 = ZZ

o2 : ZZ-module, free
i3 : cl PP3

       1
o3 = ZZ

o3 : ZZ-module, free
i4 : fromPicToCl PP3

o4 = | 1 |

              1        1
o4 : Matrix ZZ  <--- ZZ
i5 : FF7 = hirzebruchSurface 7;
i6 : pic FF7 == cl FF7

o6 = true
i7 : fromPicToCl FF7

o7 = | 1 0 |
     | 0 1 |

              2        2
o7 : Matrix ZZ  <--- ZZ
For weighted projective space, the inclusion corresponds to l ℤ in ℤ, where l = lcm(q0,…, qd ).
i8 : X = weightedProjectiveSpace {1,2,3};
i9 : pic X

       1
o9 = ZZ

o9 : ZZ-module, free
i10 : cl X

        1
o10 = ZZ

o10 : ZZ-module, free
i11 : fromPicToCl X

o11 = | -6 |

               1        1
o11 : Matrix ZZ  <--- ZZ
i12 : Y = weightedProjectiveSpace {1,2,2,3,4};
i13 : pic Y

        1
o13 = ZZ

o13 : ZZ-module, free
i14 : cl Y

        1
o14 = ZZ

o14 : ZZ-module, free
i15 : fromPicToCl Y

o15 = | -12 |

               1        1
o15 : Matrix ZZ  <--- ZZ
The following examples illustrate some other possibilities.
i16 : C = normalToricVariety({{1,0,0},{0,1,0},{0,0,1},{1,1,-1}},{{0,1,2,3}});
i17 : pic C

o17 = 0

o17 : ZZ-module
i18 : cl C

        1
o18 = ZZ

o18 : ZZ-module, free
i19 : fromPicToCl C

o19 = 0

               1
o19 : Matrix ZZ  <--- 0
i20 : X = normalToricVariety(id_(ZZ^3) | - id_(ZZ^3));
i21 : rays X

o21 = {{1, 1, 1}, {-1, 1, 1}, {1, -1, 1}, {-1, -1, 1}, {1, 1, -1}, {-1, 1,
      -----------------------------------------------------------------------
      -1}, {1, -1, -1}, {-1, -1, -1}}

o21 : List
i22 : max X

o22 = {{0, 1, 2, 3}, {0, 1, 4, 5}, {0, 2, 4, 6}, {1, 3, 5, 7}, {2, 3, 6, 7},
      -----------------------------------------------------------------------
      {4, 5, 6, 7}}

o22 : List
i23 : pic X

        1
o23 = ZZ

o23 : ZZ-module, free
i24 : cl X

o24 = cokernel | 2 0 |
               | 0 2 |
               | 0 0 |
               | 0 0 |
               | 0 0 |
               | 0 0 |
               | 0 0 |

                               7
o24 : ZZ-module, quotient of ZZ
i25 : fromPicToCl X

o25 = | 0  |
      | 0  |
      | -2 |
      | 2  |
      | -2 |
      | -4 |
      | -2 |

o25 : Matrix
i26 : prune cokernel fromPicToCl X

o26 = cokernel | 2 0 0 |
               | 0 2 0 |
               | 0 0 2 |
               | 0 0 0 |
               | 0 0 0 |
               | 0 0 0 |
               | 0 0 0 |

                               7
o26 : ZZ-module, quotient of ZZ

See also

Ways to use fromPicToCl :