In this note, we show that for a smooth algebraic variety Y and a smooth m-secant section X of the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {P}^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-bundle <Equation ID="Equ8"> <EquationSource Format="TEX">$ f : \mathbb {P}(\mathcal {O}_Y \oplus \mathcal {O}_Y(E)) \longrightarrow Y, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>f</mi> <mo>:</mo> <mi mathvariant="double-struck">P</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">O</mi> <mi>Y</mi> </msub> <mo>⊕</mo> <msub> <mi mathvariant="script">O</mi> <mi>Y</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">⟶</mo> <mi>Y</mi> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where E is an effective divisor on Y satisfying <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H^1(Y, \mathcal {O}_Y(kE)) = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <msub> <mi mathvariant="script">O</mi> <mi>Y</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$k = 1, \ldots , m-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the Tschirnhausen module of the induced covering <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ f|_X : X \longrightarrow Y $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mi>f</mi> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> </msub> <mo>:</mo> <mi>X</mi> <mo stretchy="false">⟶</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> 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

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2026年1月17日

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Youngook Choi,
Hristo Iliev,
Seonja Kim

摘要

In this note, we show that for a smooth algebraic variety Y and a smooth m-secant section X of the $\mathbb {P}^1$ $P^{1}$ -bundle $ f : \mathbb {P}(\mathcal {O}_Y \oplus \mathcal {O}_Y(E)) \longrightarrow Y, $ $f : P (O_{Y} \oplus O_{Y} (E)) ⟶ Y,$ where E is an effective divisor on Y satisfying $H^1(Y, \mathcal {O}_Y(kE)) = 0$ $H^{1} (Y, O_{Y} (k E)) = 0$ for all $k = 1, \ldots , m-1$ $k = 1, \dots, m - 1$ , the Tschirnhausen module of the induced covering $ f|_X : X \longrightarrow Y $ ${f |}_{X} : X ⟶ Y$ 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

摘要