The a-th Hirzebruch surface is a smooth projective normal toric variety. It can be defined as the ℙ1-bundle over X = ℙ1 associated to the sheaf OX(0) ⊕OX(a). It is also the quotient of affine 4-space by a rank two torus.
i1 : FF3 = hirzebruchSurface 3; |
i2 : rays FF3 o2 = {{1, 0}, {0, 1}, {-1, 3}, {0, -1}} o2 : List |
i3 : max FF3 o3 = {{0, 1}, {0, 3}, {1, 2}, {2, 3}} o3 : List |
i4 : dim FF3 o4 = 2 |
i5 : ring FF3 o5 = QQ[x , x , x , x ] 0 1 2 3 o5 : PolynomialRing |
i6 : degrees ring FF3 o6 = {{1, 0}, {-3, 1}, {1, 0}, {0, 1}} o6 : List |
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 |
i8 : assert(isProjective FF3 and isSmooth FF3) |
When a = 0, we obtain ℙ1 ×ℙ1.
i9 : FF0 = hirzebruchSurface(0, CoefficientRing => ZZ/32003, Variable => y); |
i10 : rays FF0 o10 = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}} o10 : List |
i11 : max FF0 o11 = {{0, 1}, {0, 3}, {1, 2}, {2, 3}} o11 : List |
i12 : ring FF0 ZZ o12 = -----[y , y , y , y ] 32003 0 1 2 3 o12 : PolynomialRing |
i13 : degrees ring FF0 o13 = {{1, 0}, {0, 1}, {1, 0}, {0, 1}} o13 : List |
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 |
i15 : decompose I o15 = {ideal (y , y ), ideal (y , y )} 2 0 3 1 o15 : List |
i16 : assert(isProjective FF3 and isSmooth FF3) |
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.
i17 : nefGenerators FF3 o17 = | 1 0 | | 0 1 | 2 2 o17 : Matrix ZZ <--- ZZ |
i18 : nefGenerators FF0 o18 = | 1 0 | | 0 1 | 2 2 o18 : Matrix ZZ <--- ZZ |