Peter Kleinschmidt constructs (up to isomorphism) all smooth normal toric varieties with dimension $d$ and $d+2$ rays; see Kleinschmidt's "A classification of toric varieties with few generators" Aequationes Mathematicae, 35 (1998) 254-266.
When $d = 2$, we obtain a variety isomorphic to a Hirzebruch surface. By permuting the indexing of the rays and taking an automorphism of the lattice, we produce an explicit isomorphism.
i1 : X = kleinschmidt (2,{3});
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i2 : rays X
o2 = {{-1, 0}, {1, 0}, {0, 1}, {3, -1}}
o2 : List
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i3 : max X
o3 = {{0, 2}, {0, 3}, {1, 2}, {1, 3}}
o3 : List
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i4 : FF3 = hirzebruchSurface 3; |
i5 : rays FF3
o5 = {{1, 0}, {0, 1}, {-1, 3}, {0, -1}}
o5 : List
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i6 : max FF3
o6 = {{0, 1}, {0, 3}, {1, 2}, {2, 3}}
o6 : List
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i7 : permutingRays = matrix {{0,0,0,1},{0,1,0,0},{1,0,0,0},{0,0,1,0}}
o7 = | 0 0 0 1 |
| 0 1 0 0 |
| 1 0 0 0 |
| 0 0 1 0 |
4 4
o7 : Matrix ZZ <--- ZZ
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i8 : latticeAutomorphism = matrix {{0,1},{1,0}}
o8 = | 0 1 |
| 1 0 |
2 2
o8 : Matrix ZZ <--- ZZ
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i9 : assert (latticeAutomorphism * (matrix transpose rays X) * permutingRays == matrix transpose rays FF3) |
The normal toric variety associated to the pair $(d,a)$ is Fano if and only if $\sum_{i=0}^{r-1} a_i < d-r+1$.
i10 : X1 = kleinschmidt (3, {0,1});
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i11 : isFano X1 o11 = true |
i12 : X2 = kleinschmidt (4, {0,0});
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i13 : isFano X2 o13 = true |
i14 : ring X2
o14 = QQ[x ..x ]
0 5
o14 : PolynomialRing
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i15 : X3 = kleinschmidt (9, {1,2,3}, CoefficientRing => ZZ/32003, Variable => y);
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i16 : isFano X3 o16 = true |
i17 : ring X3
ZZ
o17 = -----[y ..y ]
32003 0 10
o17 : PolynomialRing
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The map from the group of torus-invariant Weil divisors to the class group is chosen so that the positive orthant corresponds to the cone of nef line bundles.
i18 : nefGenerators X
o18 = | 1 0 |
| 0 1 |
2 2
o18 : Matrix ZZ <--- ZZ
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i19 : nefGenerators X1
o19 = | 1 0 |
| 0 1 |
2 2
o19 : Matrix ZZ <--- ZZ
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i20 : nefGenerators X2
o20 = | 1 0 |
| 0 1 |
2 2
o20 : Matrix ZZ <--- ZZ
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i21 : nefGenerators X3
o21 = | 1 0 |
| 0 1 |
2 2
o21 : Matrix ZZ <--- ZZ
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