picardGroup(NormalToricVariety) -- make the Picard group

Synopsis

Function: picardGroup
Usage:

picardGroup X
Inputs:
- X, a normal toric variety,
Outputs:
- a module, a finitely generated abelian group

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. For more information, see Theorem 4.2.1 in Cox-Little-Schenck's Toric Varieties.

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

i1 : PP3 = toricProjectiveSpace 3;

i2 : assert (isSmooth PP3 and isProjective PP3)

i3 : picardGroup PP3

       1
o3 = ZZ

o3 : ZZ-module, free

i4 : assert (picardGroup PP3 === classGroup PP3 and isFreeModule picardGroup PP3)

i5 : X = smoothFanoToricVariety (4,90);

i6 : assert (isSmooth X and isProjective X and isFano X)

i7 : picardGroup X

       5
o7 = ZZ

o7 : ZZ-module, free

i8 : assert (fromCDivToPic X === fromWDivToCl X and isFreeModule picardGroup X)

i9 : U = normalToricVariety ({{4,-1},{0,1}},{{0},{1}});

i10 : assert (isSmooth U and not isComplete U and # max U =!= 1)

i11 : picardGroup U

o11 = cokernel | 4 |

                               1
o11 : ZZ-module, quotient of ZZ

i12 : assert (classGroup U	=== picardGroup U and not isFreeModule picardGroup U)

For an affine toric variety, the Picard group is trivial.

i13 : AA3 = affineSpace 3

o13 = AA3

o13 : NormalToricVariety

i14 : assert (isSimplicial AA3 and isSmooth AA3 and # max AA3 === 1)

i15 : picardGroup AA3

o15 = 0

o15 : ZZ-module

i16 : assert (picardGroup AA3 == 0 and isFreeModule picardGroup AA3)

i17 : Q = normalToricVariety ({{1,0,0},{0,1,0},{0,0,1},{1,1,-1}},{{0,1,2,3}});

i18 : assert (not isSimplicial Q and not isComplete Q and # max Q === 1)

i19 : picardGroup Q

o19 = 0

o19 : ZZ-module

i20 : assert (picardGroup Q == 0 and isFreeModule picardGroup Q)

If the fan associated to $X$ contains a cone of dimension $dim(X)$, then the Picard group is free.

i21 : Y = normalToricVariety (id_(ZZ^3) | -id_(ZZ^3));

i22 : assert (not isSimplicial Y and isProjective Y)

i23 : picardGroup Y

        1
o23 = ZZ

o23 : ZZ-module, free

i24 : assert (rank picardGroup Y === 1 and isFreeModule picardGroup Y)

To avoid duplicate computations, the attribute is cached in the normal toric variety.

Ways to use this method: