Elliptic surface

Remarks

Many authors also require that $f$ be relatively minimal, meaning that no fibre contains a smooth rational curve of self-intersection $-1$, and some require a section of $f$. Example: for an elliptic curve $E$ and a smooth projective curve $C$, the projection $E \times C \to C$ has every fibre isomorphic to $E$, so it is an elliptic fibration. Example: blowing up $\mathbb{P}^2$ at the nine base points of a general pencil of plane cubics gives a rational elliptic surface, whose fibres are the cubics of the pencil. Non-example: the projection $\mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^1$ has all fibres of genus $0$, so it fails clause 2. In characteristics $2$ and $3$ one also meets quasi-elliptic fibrations, whose general fibre is a cuspidal rational curve.