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arXiv · 2509.11227

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

Youngook Choi·Hristo Iliev·Seonja Kim
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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 $k = 1, \ldots, m-1$, 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.

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Youngook Choi, Hristo Iliev, Seonja Kim. 2025-09-14. Decomposition of the Tschirnhausen module for coverings on decomposable $\mathbb{P}^1$-bundles. https://arxiv.org/abs/2509.11227

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