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NormalToricVarieties :: fromCDivToPic

fromCDivToPic -- get the map from Cartier divisors to 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. Hence, there is a surjective map from the group of torus-invariant Cartier divisors to the Picard group. This function returns a matrix representing this map.

On a smooth normal toric variety, the map from the torus-invariant Cartier divisors to the Picard group is the same as the map from the Weil divisors to the class group.

i1 : PP2 = projectiveSpace 2;
i2 : eta = fromCDivToPic PP2

o2 = | 1 1 1 |

              1        3
o2 : Matrix ZZ  <--- ZZ
i3 : eta == fromWDivToCl PP2

o3 = true
i4 : FF1 = hirzebruchSurface 1;
i5 : xi = fromCDivToPic FF1

o5 = | 1 -1 1 0 |
     | 0 1  0 1 |

              2        4
o5 : Matrix ZZ  <--- ZZ
i6 : xi == fromWDivToCl FF1

o6 = true
In general, there is a commutative diagram relating the map from the group of torus-invariant Cartier divisors to the Picard group and the map from the group of torus-invariant Weil divisors to the class group.
i7 : C = normalToricVariety({{1,0,0},{0,1,0},{0,0,1},{1,1,-1}},{{0,1,2,3}});
i8 : fromCDivToPic C

o8 = 0

                     3
o8 : Matrix 0 <--- ZZ
i9 : pic C         

o9 = 0

o9 : ZZ-module
i10 : fromWDivToCl C

o10 = | -1 -1 1 1 |

               1        4
o10 : Matrix ZZ  <--- ZZ
i11 : fromCDivToWDiv C

o11 = | 1 0 0  |
      | 0 1 0  |
      | 0 0 1  |
      | 1 1 -1 |

               4        3
o11 : Matrix ZZ  <--- ZZ
i12 : fromPicToCl C

o12 = 0

               1
o12 : Matrix ZZ  <--- 0
i13 : fromWDivToCl C * fromCDivToWDiv C == fromPicToCl C * fromCDivToPic C

o13 = true
i14 : X = normalToricVariety(id_(ZZ^3) | -id_(ZZ^3));
i15 : fromCDivToPic X

o15 = | 0 1 0 0 |

               1        4
o15 : Matrix ZZ  <--- ZZ
i16 : pic X

        1
o16 = ZZ

o16 : ZZ-module, free
i17 : fromWDivToCl X

o17 = | 1  0 0 0 0 1  0  0 |
      | 1  0 0 0 0 0  1  0 |
      | 0  0 1 0 0 1  0  0 |
      | -1 0 0 0 1 -1 -1 0 |
      | 0  1 0 0 0 0  1  0 |
      | 1  0 0 1 0 1  1  0 |
      | 1  0 0 0 0 0  0  1 |

o17 : Matrix
i18 : fromCDivToWDiv X

o18 = | 1  -3 1  2  |
      | 1  -3 1  0  |
      | -1 -1 1  2  |
      | -1 -1 1  0  |
      | 1  -1 -1 0  |
      | 1  -1 -1 -2 |
      | -1 1  -1 0  |
      | -1 1  -1 -2 |

               8        4
o18 : Matrix ZZ  <--- ZZ
i19 : fromPicToCl X

o19 = | 0  |
      | 0  |
      | -2 |
      | 2  |
      | -2 |
      | -4 |
      | -2 |

o19 : Matrix
i20 : fromWDivToCl X * fromCDivToWDiv X == fromPicToCl X * fromCDivToPic X

o20 = true

See also

Ways to use fromCDivToPic :