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

## Synopsis

• Function: isFano
• Usage:
isFano X
• Inputs:
• X, ,
• 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 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.