I learned some basic properties of Hirzebruch surface mainly from Vakil's notes "the rising sea", section 20.2.9. the Hirzebruch surface is defined as $\mathbb{F}_n:=\operatorname{Proj} (\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n) )$, and its Picard group is isomorphic to $\mathbb{Z}^2$. Moreover, we can easily derive the nef cone and the effective cone, and draw a picture of it:
Now I want to know which line bundle is the very ample line bundle, and mark it in the picture. Is there any results about this? Is there any algorithm to do this? Is there any databases/atlas/references for that? Thanks a lot!
What I've known about:(restricted to Hirzebruch surface $\mathbb{F}_n$, $\mathcal{L},\mathcal{U}$ be line bundles on $\mathbb{F}_n$)
- Every very ample bundle is ample, so contained in the interior of the nef cone.
- If $\mathcal{L}$ is very ample, and $\mathcal{U}$ is generated by global sections, then $\mathcal{L}\otimes \mathcal{U}$ is very ample.(see:https://math.stackexchange.com/questions/86202)
- the line bundles corresponding to $h$ and $f$ are generated by global sections, but the line bundles corresponding to $b$ is not.(I guess so)
- There is a database about the algebraic surface: https://superficie.info/ but most informations are only the numerical invariants, but not about very ample bundles.
- We have the criterian for the global-generated line bundle $\mathcal{L}$ to be very ample: let $f: \mathbb{F}_n \longrightarrow \mathbb{P}^m$ be a projective morphism induced by $|\mathcal{L}|$. If it is injective on closed points and injective on tangent vectors at closed points, then $f$ is a closed embedding. But I don't know how to use this criterian in algebraic surfaces.
