Let $F_0$ and $F_1$ be general homogeneous cubic forms with base locus $B$ of nine points. The incidence variety
$$
E(1)=\{([z],[s:t])\in\mathbb{CP}^2\times\mathbb{CP}^1:
sF_0(z)+tF_1(z)=0\}
$$
is the blowup of $\mathbb{CP}^2$ at $B$. Projection onto the second factor is the <elliptic fibration of the rational elliptic surface>
$$
\pi:E(1)\longrightarrow\mathbb{CP}^1,
\qquad
([z],[s:t])\longmapsto[s:t],
$$
and its fibers are connected plane cubics. Over any base point $p\in B$, the exceptional curve $E_p=\{p\}\times\mathbb{CP}^1$ maps isomorphically to the base, so it is a holomorphic section.

Write $H$ for the pullback of a line and $E_1,\ldots,E_9$ for the exceptional classes. The fiber class and the <canonical class of the rational elliptic surface> are
$$
F=3H-\sum_{i=1}^9E_i,
\qquad
K_{E(1)}=-3H+\sum_{i=1}^9E_i=-F.
$$
The degree of $\pi|_C$ is $C\cdot F=d$. The <Adjunction formula> now yields
$$
2g(C)-2=C^2+K_{E(1)}\cdot C=k-d.
$$
Therefore the <genus formula for a multisection of the rational elliptic surface> is
$$
g(C)=1+\frac{k-d}{2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 146 / 4 / c / Solution

{title2=$\pi:E(1)\to\mathbb{CP}^1$}

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.

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 Incoming links: Elliptic fibration of the rational elliptic surface