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weilToCartier -- the line bundle given by a Cartier divisor

Description

L must a list of weights, exactly one for each ray of the fan. Then the list of weights for each ray describes a Weil divisor on the toric variety. If the Weil divisor defined by these weights defines in fact a Cartier divisor, then weilToCartier computes the toric vector bundle associated to the Cartier divisor.

If no further options are given then the resulting bundle will be in Klyachko's description: 
i1 : F = hirzebruchFan 3

o1 = {ambient dimension => 2         }
      number of generating cones => 4
      number of rays => 4
      top dimension of the cones => 2

o1 : Fan
 
i2 : E =weilToCartier({1,-3,4,-2},F)

o2 = {dimension of the variety => 2 }
      number of affine charts => 4
      number of rays => 4
      rank of the vector bundle => 1

o2 : ToricVectorBundleKlyachko
 
i3 : details E

o3 = HashTable{| -1 | => (| 1 |, | -4 |)}
               | 3  |
               | 0  | => (| 1 |, | -1 |)
               | -1 |
               | 0 | => (| 1 |, | 3 |)
               | 1 |
               | 1 | => (| 1 |, | 2 |)
               | 0 |

o3 : HashTable

If the option "Type" => "Kaneyama" is given then the resulting bundle will be in Kaneyama's description: 
i4 : F = hirzebruchFan 3

o4 = {ambient dimension => 2         }
      number of generating cones => 4
      number of rays => 4
      top dimension of the cones => 2

o4 : Fan
 
i5 : E =weilToCartier({1,-3,4,-2},F,"Type" => "Kaneyama")

o5 = {dimension of the variety => 2 }
      number of affine charts => 4
      rank of the vector bundle => 1

o5 : ToricVectorBundleKaneyama
 
i6 : details E

o6 = (HashTable{0 => (| 0 -1 |, | 13 |)}, HashTable{(0, 1) => | 1 |})
                      | 1 3  |  | 3  |              (0, 2) => | 1 |
                1 => (| 0  -1 |, | 7 |)             (1, 3) => | 1 |
                      | -1 3  |  | 1 |              (2, 3) => | 1 |
                2 => (| 1 0 |, | 2 |)
                      | 0 1 |  | 3 |
                3 => (| 1 0  |, | 2 |)
                      | 0 -1 |  | 1 |

o6 : Sequence

See also

Ways to use weilToCartier:

  • weilToCartier(List,Fan)

For the programmer

The object weilToCartier is a method function with options.

The source of this document is in OldToricVectorBundles.m2:3834:0.