Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f: X \rightarrow B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> be a relatively minimal hyperelliptic fibration of genus g. For such a fibration f, Xiao introduced a series of singularity indices <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$s_i(f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>s</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$2\le i \le g+2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>g</mi> <mo>+</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. These indices provide an effective way to study the geometry of f. It is known that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$s_i(f)\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>s</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$i\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, but it is not clear whether <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$s_2(f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>s</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is non-negative. In this note, we construct a sequence of hyperelliptic fibrations with <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$s_2(f)<0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>s</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, where the genus g can be arbitrarily large.

A note on singularity indices of hyperelliptic fibrations

期刊论文
发表日期：

2025年12月1日

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Cheng Gong,
Zhiming Guo,
Xin Lü

摘要

Let $f: X \rightarrow B$ $f : X \to B$ be a relatively minimal hyperelliptic fibration of genus g. For such a fibration f, Xiao introduced a series of singularity indices $s_i(f)$ $s_{i} (f)$ for $2\le i \le g+2$ $2 \leq i \leq g + 2$ . These indices provide an effective way to study the geometry of f. It is known that $s_i(f)\ge 0$ $s_{i} (f) \geq 0$ for $i\ge 3$ $i \geq 3$ , but it is not clear whether $s_2(f)$ $s_{2} (f)$ is non-negative. In this note, we construct a sequence of hyperelliptic fibrations with $s_2(f)<0$ $s_{2} (f) < 0$ , where the genus g can be arbitrarily large.