- Usage:
`isSimplicial X`

- Function: isSimplicial
- Inputs:
`X`, a normal toric variety

- Outputs:
- a Boolean value, that is true if the minimal nonzero lattice points on the rays in each maximal cone in the associated fan of form part of a
*ℚ*-basis

- a Boolean value, that is true if the minimal nonzero lattice points on the rays in each maximal cone in the associated fan of form part of a

A normal toric variety is simplicial 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 torus-invariant Weil divisor on
*X*has a positive integer multiple that 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 : PP1 = toricProjectiveSpace 1; |

i2 : assert (isSimplicial PP1 and isProjective PP1) |

i3 : FF7 = hirzebruchSurface 7; |

i4 : assert (isSimplicial FF7 and isProjective FF7) |

i5 : AA3 = affineSpace 3; |

i6 : assert (isSimplicial AA3 and not isComplete AA3 and # max AA3 === 1) |

i7 : P12234 = weightedProjectiveSpace {1,2,2,3,4}; |

i8 : assert (isSimplicial P12234 and isProjective P12234) |

i9 : U = normalToricVariety ({{4,-1},{0,1}},{{0,1}}); |

i10 : assert (isSimplicial U and not isSmooth U) |

However, not all normal toric varieties are simplicial.

i11 : Q = normalToricVariety ({{1,0,0},{0,1,0},{0,0,1},{1,1,-1}},{{0,1,2,3}}); |

i12 : assert (not isSmooth Q and not isSimplicial Q and not isComplete Q) |

i13 : Y = normalToricVariety ( id_(ZZ^3) | - id_(ZZ^3)); |

i14 : assert (not isSimplicial Y and isProjective Y) |

- Basic invariants and properties of normal toric varieties
- rays(NormalToricVariety) -- get the rays of the associated fan
- max(NormalToricVariety) -- get the maximal cones in the associated fan
- isSmooth(NormalToricVariety) -- whether a normal toric variety is smooth
- makeSimplicial(NormalToricVariety) -- make a birational simplicial toric variety