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

Youngook Choi, Hristo Iliev, Seonja Kim

TL;DR

The paper studies finite $m$-fold coverings $\varphi:X\to Y$ that arise as smooth $m$-sections of decomposable $\mathbb{P}^1$-bundles over $Y$, and proves that under a vanishing condition $H^1(Y,\mathcal{O}_Y(kE+\Delta))=0$ for $k=1,\dots,m-1$, the Tschirnhausen module $\mathcal{E}^\vee$ of the covering decomposes completely into a direct sum of line bundles. The main result shows $\varphi_*\mathcal{O}_X$ splits as $\mathcal{O}_Y\oplus\mathcal{O}_Y(-E-\Delta)\oplus\cdots\oplus\mathcal{O}_Y(-(m-1)E-\Delta)$, and correspondingly $\mathcal{E}^\vee\cong\big[\mathcal{O}_Y\oplus\mathcal{O}_Y(-E)\oplus\cdots\oplus\mathcal{O}_Y(-(m-2)E)\big]\otimes\mathcal{O}_Y(-E-\Delta)$; these decompositions are obtained via $f_*$ and Ext$^1$-vanishing, using the projection formula on the decomposable ruled surface $\mathcal{B}=\mathbb{P}(\mathcal{O}_Y\oplus\mathcal{O}_Y(E))$. In the special case of curves, the results specialize to explicit genus and degree formulas, yield Veronese-type decompositions, and connect to geometric realizations on cones over $Y_e$, with concrete corollaries for curves in $|mH|$ and $|mH+q|$, including a plane-curve example. The work extends prior cases (notably m=3) and supplies tools for constructing components of Hilbert schemes of curves via $m$-covers.

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.

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

TL;DR

The paper studies finite -fold coverings that arise as smooth -sections of decomposable -bundles over , and proves that under a vanishing condition for , the Tschirnhausen module of the covering decomposes completely into a direct sum of line bundles. The main result shows splits as , and correspondingly ; these decompositions are obtained via and Ext-vanishing, using the projection formula on the decomposable ruled surface . In the special case of curves, the results specialize to explicit genus and degree formulas, yield Veronese-type decompositions, and connect to geometric realizations on cones over , with concrete corollaries for curves in and , including a plane-curve example. The work extends prior cases (notably m=3) and supplies tools for constructing components of Hilbert schemes of curves via -covers.

Abstract

In this note, we show that for a smooth algebraic variety and a smooth -section of the -bundle where is an effective divisor on satisfying for all , the Tschirnhausen module of the induced covering is completely decomposable. We then apply it to coverings of curves arising in such a way.

Paper Structure

This paper contains 3 sections, 8 theorems, 49 equations.

Table of Contents

  1. Introduction and statement
  2. Proof of Theorem \ref{['TheoremM']}
  3. The case of curves

Key Result

Theorem 1

Let $Y$ be a smooth projective variety of dimension $n \geq 1$, and let $E$ and $\Delta$ be effective divisors on $Y$ such that where $m \geq 2$. Consider the projective bundle $\mathcal{B} := \mathbb{P}(\mathcal{O}_Y \oplus \mathcal{O}_Y(E)),$ following Hartshorne's convention, with projection $f : \mathcal{B} \to Y.$ Let $H$ denote the tautological divisor on $\mathcal{B}$, so that $\mathcal{O}

Theorems & Definitions (18)

  • Theorem 1
  • Proposition 2
  • proof
  • proof : Proof of Theorem \ref{['TheoremM']}
  • Proposition 3
  • proof
  • Corollary 4
  • proof
  • Remark 5
  • Corollary 6
  • ...and 8 more