next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
NormalToricVarieties :: isSimplicial(NormalToricVariety)

isSimplicial(NormalToricVariety) -- whether a toric variety is simplicial

Synopsis

Description

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 . 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 -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.
i1 : isSimplicial projectiveSpace 4

o1 = true
i2 : isSimplicial weightedProjectiveSpace {1,2,3}

o2 = true
i3 : isSimplicial hirzebruchSurface 7

o3 = true
However, not all normal toric varieties are simplicial.
i4 : U = normalToricVariety({{4,-1},{0,1}},{{0,1}});
i5 : isSimplicial U

o5 = true
i6 : isSmooth U

o6 = false
i7 : C = normalToricVariety({{1,0,0},{0,1,0},{0,0,1},{1,1,-1}},{{0,1,2,3}});
i8 : isSimplicial C

o8 = false

See also