The $a$-{th} Hirzebruch surface is a smooth projective normal toric variety. It can be defined as the $\PP^1$-bundle over $X = \PP^1$ associated to the sheaf ${\mathcal O}_X(0) \oplus {\mathcal O}_X(a)$. 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 ]
0 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 ]
0 3
|
i8 : assert (isWellDefined FF3 and isProjective FF3 and isSmooth FF3) |
When $a = 0$, we obtain $\PP^1 \times \PP^1$.
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 ]
32003 0 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 ]
32003 0 3
|
i15 : decompose I
o15 = {ideal (y , y ), ideal (y , y )}
2 0 3 1
o15 : List
|
i16 : assert (isWellDefined FF0 and isProjective FF3 and isSmooth FF3) |
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.
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
|