The Hirzebruch surface F_a from its fan

(The fan)

Take $N = \ZZ^2$ with rays \[ u_1 = (1,0), \quad u_2 = (0,1), \quad u_3 = (-1,a), \quad u_4 = (0,-1) \] and the four cones spanned by consecutive pairs. Each consecutive pair has determinant $\pm 1$: \[ \det(u_1, u_2) = 1, \quad \det(u_2, u_3) = 1, \quad \det(u_3, u_4) = 1, \quad \det(u_4, u_1) = 1 , \] so $\FF_a$ is smooth, and the support is all of $N_\RR$, so it is complete.

(Intersection numbers)

Read the relations $u_{i-1} + u_{i+1} = a_i u_i$ off the coordinates: \[ u_1 + u_3 = (0,a) = a \cdot u_2 , \qquad u_3 + u_1 = (0,a) = -a \cdot u_4 , \qquad u_4 + u_2 = (0,0) = 0 \cdot u_1 . \] Hence \[ D_2^2 = -a, \qquad D_4^2 = a, \qquad D_1^2 = D_3^2 = 0 . \] $D_2$ is the negative section, $D_4$ a positive section, $D_1$ and $D_3$ the two invariant fibres of $\FF_a \to \PP^1$, which is the toric morphism induced by projection $N \to \ZZ$ onto the first coordinate.

(Class group)

$\Cl(\FF_a) = \ZZ^4 / M \cong \ZZ^2$ of rank $4 - 2 = 2$, generated by a fibre $F = D_1$ and the section $S = D_2$, with $F^2 = 0$, $F \cdot S = 1$, $S^2 = -a$. The fan is smooth so $\Pic = \Cl$ and $\rank \Pic(\FF_a) = 2$. The remaining classes are $D_3 \sim F$ and $D_4 \sim S + aF$. Also $-K = \sum D_i \sim 2S + (a+2)F$.

$\FF_0$ has rays $(\pm 1, 0), (0, \pm 1)$, a product fan, so $\FF_0 = \PP^1 \times \PP^1$, the smooth quadric in $\PP^3$. $\FF_1$ is $\PP^2$ blown up at a point, with $D_2$ the exceptional curve of self-intersection $-1$. For $a \geq 2$ the surface is not Fano: $-K \cdot D_2 = 2 - a \leq 0$. Every $\FF_a$ is smooth, for all $a \geq 0$, since each maximal cone has determinant $\pm 1$.

(The Segre embedding of $\FF_0$)

The Segre embedding $\PP^1 \times \PP^1 \to \PP^3$, $([a:b],[c:d]) \mapsto [ac : ad : bc : bd]$, has image $V(xw - yz)$. Its four monomials have exponent vectors $(1,0,1,0)$, $(1,0,0,1)$, $(0,1,1,0)$, $(0,1,0,1)$ in the variables $a, b, c, d$, the vertices of a square, and the affine cone over the image is the affine toric variety $V(xw - yz) \subseteq \CC^4$ of the cone over that square.

Write down the fan of the Hirzebruch surface F_a and compute its class group and intersection numbers.

Why is F_0 the quadric surface?

What is a fan for the total space of $\OO_{\PP^1}(n)$?

Describe the Segre embedding $\PP^1 \times \PP^1 \to \PP^3$ and its polytope.

Dependencies

T-TORSURFSmooth complete toric surfaces and their intersection numbersuses

Backlinks

FE-FULHIRZBUNThe Hirzebruch surface as a projective bundle over the lineuses

What links to this

Toric surfaces and toric morphisms