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

- Function: isFano
- Usage:
    `isFano X`
- Inputs:
  - X, a normal toric variety,
- Outputs:
  - a Boolean value, 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.

## See also

- finding attributes and properties -- information about accessing features of a normal toric variety
- 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

## Ways to use this method:

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NormalToricVarieties/DivisorsDocumentation.m2:1947:0.