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NormalToricVarieties :: isFano

isFano -- whether a normal toric variety is Fano

Synopsis

Description

A normal toric variety is Fano if its anticanonical divisor, namely the sum of all the torus-invariant prime 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 = projectiveSpace 3;
i2 : isFano PP3

o2 = true
i3 : K = toricDivisor PP3

o3 = - D  - D  - D  - D
        0    1    2    3

o3 : ToricDivisor on PP3
i4 : isAmple (-K)

o4 = true
i5 : apply(5, d -> isFano projectiveSpace (d+1))

o5 = {true, true, true, true, true}

o5 : List
There are eighteen smooth Fano toric threefolds.
i6 : all(18, i -> (X := smoothFanoToricVariety(3,i); isSmooth X and isFano X))

o6 = true
There are also many singular Fano toric varieties
i7 : X = normalToricVariety matrix {{1,0,-1},{0,1,-1}};
i8 : isSmooth X

o8 = false
i9 : isFano X

o9 = true
i10 : Y = normalToricVariety matrix {{1,1,-1,-1},{0,1,1,-1}}

o10 = Y

o10 : NormalToricVariety
i11 : isSmooth Y

o11 = false
i12 : isFano Y

o12 = true
i13 : Z = normalToricVariety(id_(ZZ^3) | -id_(ZZ^3));
i14 : isSmooth Z

o14 = false
i15 : isFano Z

o15 = true

See also

Ways to use isFano :