A normal toric variety is smooth if every cone in its fan is smooth and a cone is smooth if its minimal generators are linearly independent over
ℤ. In fact, the following conditions on a normal toric variety
X are equivalent:
- X is smooth;
- every Weil divisor on X is Cartier;
- the Picard group of X equals the class group of X;
- X has no singularities.
Projective spaces and Hirzebruch surfaces are smooth.
isSmooth projectiveSpace 4 |
isSmooth hirzebruchSurface 7 |
However, not all normal toric varieties are smooth.
isSmooth weightedProjectiveSpace {1,2,3} |
U = normalToricVariety({{4,-1},{0,1}},{{0,1}}); |
isSimplicial U |
isSmooth U |
U' = normalToricVariety({{4,-1},{0,1}},{{0},{1}}); |
isSmooth U' |