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;
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i2 : isFano PP3
o2 = true
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i3 : K = toricDivisor PP3
o3 = - D - D - D - D
0 1 2 3
o3 : ToricDivisor on PP3
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i4 : isAmple (-K)
o4 = true
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i5 : apply(5, d -> isFano projectiveSpace (d+1))
o5 = {true, true, true, true, true}
o5 : List
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There are eighteen smooth Fano toric threefolds.
i6 : all(18, i -> (X := smoothFanoToricVariety(3,i); isSmooth X and isFano X))
o6 = true
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There are also many singular Fano toric varieties
i7 : X = normalToricVariety matrix {{1,0,-1},{0,1,-1}};
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i8 : isSmooth X
o8 = false
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i9 : isFano X
o9 = true
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i10 : Y = normalToricVariety matrix {{1,1,-1,-1},{0,1,1,-1}}
o10 = Y
o10 : NormalToricVariety
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i11 : isSmooth Y
o11 = false
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i12 : isFano Y
o12 = true
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i13 : Z = normalToricVariety(id_(ZZ^3) | -id_(ZZ^3));
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i14 : isSmooth Z
o14 = false
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i15 : isFano Z
o15 = true
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