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.
i1 : isSmooth projectiveSpace 4
o1 = true
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i2 : isSmooth hirzebruchSurface 7
o2 = true
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However, not all normal toric varieties are smooth.
i3 : isSmooth weightedProjectiveSpace {1,2,3}
o3 = false
|
i4 : U = normalToricVariety({{4,-1},{0,1}},{{0,1}});
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i5 : isSimplicial U
o5 = true
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i6 : isSmooth U
o6 = false
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i7 : U' = normalToricVariety({{4,-1},{0,1}},{{0},{1}});
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i8 : isSmooth U'
o8 = true
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