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 PP3 |
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) |
To avoid duplicate computations, the attribute is cached in the normal toric variety.