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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.

PP2 = projectiveSpace 2;
A = fromWDivToCl PP2
source A == wDiv PP2
target A == cl PP2
X = weightedProjectiveSpace {1,2,2,3,4};
fromWDivToCl X
FF7 = hirzebruchSurface 7;
A' = fromWDivToCl FF7
(source A', target A') == (wDiv FF7, cl FF7)
U = normalToricVariety({{4,-1},{0,1}},{{0,1}});
fromWDivToCl U
wDiv U
cl U
This matrix also induces the grading on the total coordinate ring of toric variety.
degrees ring PP2
degrees ring X
degrees ring FF7
The optional argument WeilToClass for the constructor normalToricVariety allows one to specify a basis of the class group.

See also

Ways to use fromWDivToCl :