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.
Best Answer
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, $$ \dim H^0(\mathbb{F}_n, \mathcal{O}(D_{a,b})) = \begin{cases} \sum_{i = 0}^{\min(a,\lfloor b/n \rfloor)} (b - in + 1), & \text{if $a \ge 0$},\\ 0, & \text{if $a < 0$}. \end{cases} $$