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NormalToricVarieties :: isSmooth(NormalToricVariety)

isSmooth(NormalToricVariety) -- whether a toric variety is smooth

Synopsis

Description

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 . 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
i2 : isSmooth hirzebruchSurface 7

o2 = true
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}});
i5 : isSimplicial U

o5 = true
i6 : isSmooth U

o6 = false
i7 : U' = normalToricVariety({{4,-1},{0,1}},{{0},{1}});
i8 : isSmooth U'

o8 = true

See also