# hirzebruchSurface(ZZ) -- make any Hirzebruch surface

## Synopsis

• Function: hirzebruchSurface
• Usage:
hirzebruchSurface a
• Inputs:
• a, an integer, that determines which Hirzebruch surface
• Optional inputs:
• CoefficientRing => a ring, default value QQ, that specifies the coefficient ring of the total coordinate ring
• Variable => , default value x, that specifies the base name for the indexed variables in the total coordinate ring
• Outputs:
• , that is a Hirzebruch surface

## Description

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