- Usage:
`isFano X`

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

- Outputs:
- a Boolean value, that is true if the normal toric variety is Fano

A normal toric variety is Fano if its anticanonical divisor, namely the sum of all the torus-invariant irreducible divisors, is ample. This is equivalent to saying that the polyhedron associated to the anticanonical divisor is a reflexive polytope.

Projective space is Fano.

i1 : PP3 = toricProjectiveSpace 3; |

i2 : assert isFano PP3 |

i3 : K = toricDivisor PP3 o3 = - PP3 - PP3 - PP3 - PP3 0 1 2 3 o3 : ToricDivisor on normalToricVariety((({{-1, -1, -1}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}(,({{0, 1, 2}, {0, 1, 3}, {0, 2, 3}, {1, 2, 3}} ))))) |

i4 : isAmple (-K) o4 = true |

i5 : assert all (5, d -> isFano toricProjectiveSpace (d+1)) |

There are eighteen smooth Fano toric threefolds.

i6 : assert all (18, i -> (X := smoothFanoToricVariety (3,i); isSmooth X and isFano X)) |

There are also many singular Fano toric varieties.

i7 : X = normalToricVariety matrix {{1,0,-1},{0,1,-1}}; |

i8 : assert (not isSmooth X and isFano X) |

i9 : Y = normalToricVariety matrix {{1,1,-1,-1},{0,1,1,-1}}; |

i10 : assert (not isSmooth Y and isFano Y) |

i11 : Z = normalToricVariety (id_(ZZ^3) | -id_(ZZ^3)); |

i12 : assert (not isSmooth Z and isFano Z) |

- Basic invariants and properties of normal toric varieties
- toricDivisor(NormalToricVariety) -- make the canonical divisor
- isAmple(ToricDivisor) -- whether a torus-invariant Weil divisor is ample
- smoothFanoToricVariety(ZZ,ZZ) -- get a smooth Fano toric variety from database