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NormalToricVarieties :: fromWDivToCl

fromWDivToCl -- get the map from Weil divisors to the class group

Synopsis

Description

For a normal toric variety, the class group has a presentation defined by the map from the group of torus-characters to group of torus-invariant Weil divisors induced by minimal nonzero lattice points on the rays of the associated fan. Hence, there is a surjective map from the group of torus-invariant Weil divisors to the class group. This method returns a matrix representing this map. Since the ordering on the rays of the toric variety determines a basis for the group of torus-invariant Weil divisors, this matrix is determined by a choice of basis for the class group.

The examples illustrate some of the possible maps from the group of torus-invariant Weil divisors to the class group.

i1 : PP2 = projectiveSpace 2;
i2 : A = fromWDivToCl PP2

o2 = | 1 1 1 |

              1        3
o2 : Matrix ZZ  <--- ZZ
i3 : source A == wDiv PP2

o3 = true
i4 : target A == cl PP2

o4 = true
i5 : X = weightedProjectiveSpace {1,2,2,3,4};
i6 : fromWDivToCl X

o6 = | 1 2 2 3 4 |

              1        5
o6 : Matrix ZZ  <--- ZZ
i7 : FF7 = hirzebruchSurface 7;
i8 : A' = fromWDivToCl FF7

o8 = | 1 -7 1 0 |
     | 0 1  0 1 |

              2        4
o8 : Matrix ZZ  <--- ZZ
i9 : (source A', target A') == (wDiv FF7, cl FF7)

o9 = true
i10 : U = normalToricVariety({{4,-1},{0,1}},{{0,1}});
i11 : fromWDivToCl U

o11 = | 1 1 |

o11 : Matrix
i12 : wDiv U

        2
o12 = ZZ

o12 : ZZ-module, free
i13 : cl U

o13 = cokernel | 4 |

                               1
o13 : ZZ-module, quotient of ZZ
This matrix also induces the grading on the total coordinate ring of toric variety.
i14 : degrees ring PP2

o14 = {{1}, {1}, {1}}

o14 : List
i15 : degrees ring X

o15 = {{1}, {2}, {2}, {3}, {4}}

o15 : List
i16 : degrees ring FF7

o16 = {{1, 0}, {-7, 1}, {1, 0}, {0, 1}}

o16 : List
The optional argument WeilToClass for the constructor normalToricVariety allows one to specify a basis of the class group.

See also

Ways to use fromWDivToCl :