# fromCDivToWDiv(NormalToricVariety) -- get the map from Cartier divisors to Weil divisors

## Synopsis

• Usage:
fromCDivToWDiv X
• Function: fromCDivToWDiv
• Inputs:
• Outputs:
• , representing the inclusion map from the group of torus-invariant Cartier divisors to the group of torus-invariant Weil divisors

## Description

The group of torus-invariant Cartier divisors is the subgroup of all locally principal torus-invariant Weil divisors. This function produces the inclusion map with respect to the chosen bases for the two finitely-generated abelian groups.

On a smooth normal toric variety, every torus-invariant Weil divisor is Cartier, so the inclusion map is simply the identity map.

 `i1 : PP2 = toricProjectiveSpace 2;` `i2 : assert (isSmooth PP2 and isProjective PP2)` ```i3 : fromCDivToWDiv PP2 o3 = | 1 0 0 | | 0 1 0 | | 0 0 1 | 3 3 o3 : Matrix ZZ <--- ZZ``` `i4 : assert (fromCDivToWDiv PP2 === id_(weilDivisorGroup PP2))`
 `i5 : X = smoothFanoToricVariety (4,20);` `i6 : assert (isSmooth X and isProjective X and isFano X)` ```i7 : fromCDivToWDiv X o7 = | 1 0 0 0 0 0 0 | | 0 1 0 0 0 0 0 | | 0 0 1 0 0 0 0 | | 0 0 0 1 0 0 0 | | 0 0 0 0 1 0 0 | | 0 0 0 0 0 1 0 | | 0 0 0 0 0 0 1 | 7 7 o7 : Matrix ZZ <--- ZZ``` `i8 : assert (fromCDivToWDiv X === id_(weilDivisorGroup X))`
 `i9 : U = normalToricVariety ({{4,-1},{0,1}},{{0},{1}});` `i10 : assert (isSmooth U and not isComplete U)` ```i11 : fromCDivToWDiv U o11 = | 1 0 | | 0 1 | 2 2 o11 : Matrix ZZ <--- ZZ``` `i12 : assert (fromCDivToWDiv U === id_(weilDivisorGroup U))`

On a simplicial normal toric variety, every torus-invariant Weil divisor is -Cartier; every torus-invariant Weil divisor has a positive integer multiple that is Cartier.

 `i13 : C = normalToricVariety ({{4,-1},{0,1}},{{0,1}});` ```i14 : fromCDivToWDiv C o14 = | 4 -1 | | 0 1 | 2 2 o14 : Matrix ZZ <--- ZZ``` ```i15 : prune cokernel fromCDivToWDiv C o15 = cokernel | 4 | 1 o15 : ZZ-module, quotient of ZZ``` `i16 : assert (rank cokernel fromCDivToWDiv C === 0)`

In general, the Cartier divisors are only a subgroup of the Weil divisors.

 `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)` ```i19 : fromCDivToWDiv Q o19 = | 1 0 0 | | 0 1 0 | | 0 0 1 | | 1 1 -1 | 4 3 o19 : Matrix ZZ <--- ZZ``` ```i20 : prune coker fromCDivToWDiv Q 1 o20 = ZZ o20 : ZZ-module, free``` `i21 : assert (rank coker fromCDivToWDiv Q === 1)`
 `i22 : Y = normalToricVariety (id_(ZZ^3) | -id_(ZZ^3));` `i23 : assert (not isSimplicial Y and isComplete Y)` ```i24 : fromCDivToWDiv Y o24 = | 1 1 1 1 | | -1 1 1 1 | | 1 -1 1 1 | | -1 -1 1 1 | | 1 1 -1 1 | | -1 1 -1 1 | | 1 -1 -1 1 | | -1 -1 -1 1 | 8 4 o24 : Matrix ZZ <--- ZZ``` ```i25 : prune cokernel fromCDivToWDiv Y o25 = cokernel | 2 0 0 | | 0 2 0 | | 0 0 2 | | 0 0 0 | | 0 0 0 | | 0 0 0 | | 0 0 0 | 7 o25 : ZZ-module, quotient of ZZ``` `i26 : assert (rank coker fromCDivToWDiv Y === 4)`