# makeSimplicial(NormalToricVariety) -- make a birational simplicial toric variety

## Description

A normal toric variety is simplicial if every cone in its fan is simplicial and a cone is simplicial if its minimal generators are linearly independent over $\QQ$. In fact, the following conditions on a normal toric variety $X$ are equivalent:

• X is simplicial;
• every Weil divisor on X has a positive integer multiple that is Cartier;
• the Picard group of X has finite index in the class group of X;
• X has only finite quotient singularities.

 i1 : X = normalToricVariety (id_(ZZ^3) | - id_(ZZ^3)); i2 : assert not isSimplicial X i3 : Y1 = makeSimplicial X; i4 : assert isSimplicial Y1 i5 : assert (rays Y1 === rays X) i6 : max Y1 o6 = {{0, 1, 3}, {0, 1, 5}, {0, 2, 3}, {0, 2, 6}, {0, 4, 5}, {0, 4, 6}, {1, ------------------------------------------------------------------------ 3, 7}, {1, 5, 7}, {2, 3, 7}, {2, 6, 7}, {4, 5, 7}, {4, 6, 7}} o6 : List i7 : max X o7 = {{0, 1, 2, 3}, {0, 1, 4, 5}, {0, 2, 4, 6}, {1, 3, 5, 7}, {2, 3, 6, 7}, ------------------------------------------------------------------------ {4, 5, 6, 7}} o7 : List i8 : Y2 = makeSimplicial(X, Strategy => 1); i9 : assert isSimplicial Y2 i10 : assert (rays Y2 === rays X) i11 : max Y2 o11 = {{0, 1, 3}, {0, 1, 5}, {0, 2, 3}, {0, 2, 6}, {0, 4, 5}, {0, 4, 6}, {1, ----------------------------------------------------------------------- 3, 7}, {1, 5, 7}, {2, 3, 7}, {2, 6, 7}, {4, 5, 7}, {4, 6, 7}} o11 : List i12 : max Y1 == max Y2 o12 = true
 i13 : PP3 = toricProjectiveSpace 3; i14 : assert isSimplicial PP3 i15 : Z = makeSimplicial PP3; i16 : assert (rays Z === rays PP3 and max Z === max PP3)