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 => a symbol, default value x, that specifies the base name for the indexed variables in the total coordinate ring
Outputs:
- a normal toric variety, 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

Ways to use this method: