Pome1F38B2F1E2A7Lemma 12.30.3.1F38B2F1E2A7 There is a direct sum decomposition
The restriction of a class $c=\rho \xi +Kf^*\eta $ to any ruling fiber is nonzero if and only if $\rho \ne 0$.
Proof. The relative cotangent sequence is
The two affine charts of a projective line give the direct images
For the last identity, a Laurent cocycle $g(z,t)\, \mathrm{d}t$ can be reduced, by subtracting forms holomorphic on the two charts, to its coefficient of $\mathrm{d}t/t$. Its residue is unchanged by $t_ i=b_{ij}t_ j$, with differentials taken relative to the base, so the local generators glue to the trivial line bundle. For functions, the two halves of a Laurent series exhaust the cocycle, which gives $R^1f_*\mathcal O_ M=0$. The direct images in degree zero follow because $\mathbb P^1$ has no nonconstant holomorphic functions and no nonzero holomorphic one-forms.
The Leray spectral sequences and the fact that the base is a curve now give
together with $H^0(M,\Omega ^1_{M/E})=0$. The cohomology sequence of the relative cotangent sequence is therefore the exact sequence
On a fiber, the class $\xi $ is represented on the two affine charts by $\mathrm{d}\log v=-\mathrm{d}t/t$, whose residue is nonzero, so $\xi $ splits the sequence. The pulled-back class $f^*\eta $ restricts to zero on every fiber. This proves both assertions. $\square$
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