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

Abstract

In this note, we show that for a smooth algebraic variety $Y$ and a smooth $m$-section $X$ of the ${\mathbb{P}}^1$-bundle \[ f : \mathbb{P}(\mathcal{O}_Y \oplus \mathcal{O}_Y(E)) \longrightarrow Y, \] 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 $f{|}_X:X⟶Y$ is completely decomposable. We then apply it to coverings of curves arising in such a way.