Consider the Hirzebruch surface $\mathbb{F}\_n = \mathbb{P}(\mathcal{O}\_{\mathbb{P}^1}\oplus \mathcal{O}\_{\mathbb{P}^1}(n))\rightarrow\mathbb{P}^1$. The Picard group of $\mathbb{F}\_n$ is generated by the class of the negative section $\Gamma$, such that $\Gamma^2 = -n$, and the class of a fiber $F$ of the morphism onto $\mathbb{P}^1$. Furthermore, $\Gamma$ and $F$ generate the effective cone of $\mathbb{F}\_n$ so that all divisors of the form $D\_{a,b} = a\Gamma + bF$ with $a,b\in \mathbb{N}$ have sections.


Does there exist a closed formula for the dimension of the vector space $H^0(\mathbb{F}\_n,D\_{a,b})$?


Thank you very much.



Cohomology of divisors on Hirzebruch surfaces
Sure. To compute $H^0$ one can first pushforward to the base $\mathbb{P}^1$. If $a \ge 0$ one obtains
$$
p\_\*\mathcal{O}(a\Gamma + bF) \cong
p\_\*\mathcal{O}(a\Gamma) \otimes \mathcal{O}(b) \cong
S^a(\mathcal{O} \oplus \mathcal{O}(-n)) \otimes \mathcal{O}(b) \cong
\bigoplus\_{i=0}^a \mathcal{O}(b-in),
$$
and if $a < 0$ the pushforward is zero. Therefore,
$$
\dim H^0(\mathbb{F}\_n, \mathcal{O}(D\_{a,b})) =
\begin{cases}
\sum\_{i = 0}^{\min(a,\lfloor b/n \rfloor)} (b - in + 1),
& \text{if $a \ge 0$},\\
0, & \text{if $a < 0$}.
\end{cases}
$$


