Peter Kleinschmidt constructs (up to isomorphism) all smooth normal toric varieties with dimension
rays; see P. Kleinschmidt, A classification of toric varieties with few generators,
(1998) 254-266.
, we obtain a variety isomorphic to a Hirzebruch surface.
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;
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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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< d-r+1.
i7 : X1 = kleinschmidt(3,{0,1});
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i8 : isFano X1
o8 = true
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i9 : X2 = kleinschmidt(4,{0,0});
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i10 : isFano X2
o10 = true
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i11 : ring X2
o11 = QQ[x , x , x , x , x , x ]
0 1 2 3 4 5
o11 : PolynomialRing
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i12 : X3 = kleinschmidt(9,{1,2,3}, CoefficientRing => ZZ/32003, Variable => y);
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i13 : isFano X3
o13 = true
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i14 : ring X3
ZZ
o14 = -----[y , y , y , y , y , y , y , y , y , y , y ]
32003 0 1 2 3 4 5 6 7 8 9 10
o14 : PolynomialRing
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