Elliptic fibration of the rational elliptic surface ($\pi :E(1)\to {\mathbb{CP}}^1$)

Projection to $[s:t]$ gives an elliptic fibration with connected cubic fibers. Each of the nine exceptional divisors maps isomorphically to ${\mathbb{CP}}^1$ and is therefore a holomorphic section.

Writing $H$ for the pullback of a line and $E_1,\dots ,E_9$ for the exceptional classes, a fiber has class $F=3H-{\sum }_i{E}_i$, while the blowup formula gives

If a smooth connected complex curve $C\subset E(1)$ satisfies $C^2=k$ and has degree $d=C\cdot F$ over the base, the Adjunction formula gives