# isFano(NormalToricVariety) -- whether a normal toric variety is Fano

## Synopsis

• Usage:
isFano X
• Function: isFano
• Inputs:
• Outputs:
• , that is true if the normal toric variety is Fano

## Description

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)`