Decomposition of the Tschirnhausen module for coverings on decomposable \texorpdfstring{$\mathbb{P}^1$}{P1}-bundles

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