Decomposition of the Tschirnhausen module for coverings on decomposable ${\mathbb{P}}^1$-bundles

In this note, we show that for a smooth algebraic variety $Y$ and a smooth $m$-section $X$ of the ${\mathbb{P}}^1$-bundle $$f:\mathbb{P}({\mathcal{O}}_Y\oplus {\mathcal{O}}_Y(E))⟶Y,$$ where $E$ is an effective divisor on $Y$ satisfying $H^1(Y,{\mathcal{O}}_Y(kE))=0$ for all $k=1,\dots ,m-1$, the Tschirnhausen module of the induced covering $f{\mathrm{\mid }}_X:X⟶Y$ is completely decomposable. We then apply it to coverings of curves arising in such a way.