A normal toric variety is simplical if every cone in its fan is simplicial and a cone is simplicial if its minimal generators are linearly independent over
ℚ. 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;
- X is ℚ-Cartier;
- the Picard group of X has finite index in the class group of X;
- X has only finite quotient singularities.
Projective spaces, weighted projective spaces and Hirzebruch surfaces are simplicial.
isSimplicial projectiveSpace 4 |
isSimplicial weightedProjectiveSpace {1,2,3} |
isSimplicial hirzebruchSurface 7 |
However, not all normal toric varieties are simplicial.
U = normalToricVariety({{4,-1},{0,1}},{{0,1}}); |
isSimplicial U |
isSmooth U |
C = normalToricVariety({{1,0,0},{0,1,0},{0,0,1},{1,1,-1}},{{0,1,2,3}}); |
isSimplicial C |