I want to describe push-forward of the relative canonical divisor of $\mathbb{F}_n:=\mathbb{P}(\mathcal{O}(n)\oplus \mathcal(O))$ via the projection $\pi:\mathbb{F}_n\to\mathbb{P}^1,$ but I don't know which is the canonical divisor $K_{\mathbb{F}_n},$ and clearly I need it, so any help to how compute or describe this divisor will be so much usefull for my.
Thank you.
The anti canonical divisor of a toric surface may be represented as a sum of all the boundary divisors, i.e. torus invariant curves corresponding to the edges of the associated Delzant Polygon. Hence $K_{X}$ is minus the sum of all boundary divisors. In the case of a Hirzebruch surface $\mathbb{F}_{n}$ the boundary divisors consist of the two sections $S_{1},S_{2}$ (with $S_{1}^{2} = n$ and $S_{2}^{2} = -n$) and two copies of a fibre of the $\mathbb{P}^{1}$-bundle say $F$ with $F^{2} = 0 $, and $F \cdot S_{i} = 1$ for $i = 1,2 $.
So in summary $$K_{X} = - 2F -S_{1} -S_{2} .$$