Hirzebruch surface is a complete normal toric variety. It can be defined as the
. It is also the quotient of affine
-space by a rank two torus.
i1 : FF3 = hirzebruchSurface 3;
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i2 : rays FF3
o2 = {{1, 0}, {0, 1}, {-1, 3}, {0, -1}}
o2 : List
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i3 : max FF3
o3 = {{0, 1}, {0, 3}, {1, 2}, {2, 3}}
o3 : List
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i4 : dim FF3
o4 = 2
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i5 : ring FF3
o5 = QQ[x , x , x , x ]
0 1 2 3
o5 : PolynomialRing
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i6 : degrees ring FF3
o6 = {{1, 0}, {-3, 1}, {1, 0}, {0, 1}}
o6 : List
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i7 : ideal FF3
o7 = ideal (x x , x x , x x , x x )
2 3 1 2 0 3 0 1
o7 : Ideal of QQ[x , x , x , x ]
0 1 2 3
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.
i8 : FF0 = hirzebruchSurface(0, CoefficientRing => ZZ/32003, Variable => y);
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i9 : rays FF0
o9 = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}}
o9 : List
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i10 : max FF0
o10 = {{0, 1}, {0, 3}, {1, 2}, {2, 3}}
o10 : List
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i11 : dim FF0
o11 = 2
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i12 : ring FF0
ZZ
o12 = -----[y , y , y , y ]
32003 0 1 2 3
o12 : PolynomialRing
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i13 : degrees ring FF0
o13 = {{1, 0}, {0, 1}, {1, 0}, {0, 1}}
o13 : List
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i14 : I = ideal FF0
o14 = ideal (y y , y y , y y , y y )
2 3 1 2 0 3 0 1
ZZ
o14 : Ideal of -----[y , y , y , y ]
32003 0 1 2 3
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i15 : decompose I
o15 = {ideal (y , y ), ideal (y , y )}
2 0 3 1
o15 : List
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The map from the torus-invariant Weil divisors to the class group is chosen so that the positive orthant corresponds to the cone of nef line bundles.