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NormalToricVarieties :: pic

pic -- make the Picard group

Synopsis

Description

The Picard group of a variety is the group of Cartier divisors divided by the subgroup of principal divisors. For a normal toric variety, the Picard group has a presentation defined by the map from the group of torus-characters to the group of torus-invariant Cartier divisors.

When toric variety is smooth, the Picard group is isomorphic to the class group.

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 : FF7 = hirzebruchSurface 7;
i5 : pic FF7 == cl FF7

o5 = true
For an affine toric variety, the Picard group is trivial.
i6 : U = normalToricVariety({{4,-1},{0,1}},{{0,1}});
i7 : pic U

o7 = 0

o7 : ZZ-module
i8 : cl U

o8 = cokernel | 4 |

                              1
o8 : ZZ-module, quotient of ZZ
i9 : U' = normalToricVariety({{4,-1},{0,1}},{{0},{1}});
i10 : pic U'

o10 = cokernel | 4 |

                               1
o10 : ZZ-module, quotient of ZZ
i11 : cl U'

o11 = cokernel | 4 |

                               1
o11 : ZZ-module, quotient of ZZ
If the fan associated to X contains a cone of dimension dim(X) then the Picard group is free.
i12 : C = normalToricVariety({{1,0,0},{0,1,0},{0,0,1},{1,1,-1}},{{0,1,2,3}});
i13 : pic C

o13 = 0

o13 : ZZ-module
i14 : cl C

        1
o14 = ZZ

o14 : ZZ-module, free
i15 : X = normalToricVariety(id_(ZZ^3) | -id_(ZZ^3));
i16 : pic X

        1
o16 = ZZ

o16 : ZZ-module, free
i17 : cl X

o17 = cokernel | 2 0 |
               | 0 2 |
               | 0 0 |
               | 0 0 |
               | 0 0 |
               | 0 0 |
               | 0 0 |

                               7
o17 : ZZ-module, quotient of ZZ

See also

Ways to use pic :