<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f:S\rightarrow B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>S</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> a locally non-trivial fibred surface with fibres of genus <i>g</i>. Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$u_f$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation> be its unitary rank, i.e. the rank of the flat unitary part in the second Fujita decomposition. We study in detail the case when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$u_f$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation> is maximal, i.e. <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$u_f=g-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>f</mi> </msub> <mo>=</mo> <mi>g</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In this case necessarily <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$g\le 6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>≤</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation>, but examples in genus 5 and 6 are not known, and conjecturally do not exist. We prove a strong slope inequality for these extremal cases. We then use this inequality, together with results on trigonal curves, to give new constraints on the case <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$g=6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>=</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$u_f=5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>. In particular, we prove that the index of the surface is always strictly positive and give strong limitations on the possible classes of the relative canonical divisor.</p>

Locally non-trivial fibred surfaces with maximal unitary rank

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2025年7月15日

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Lidia Stoppino

摘要

Let $f:S\rightarrow B$ $f : S \to B$ a locally non-trivial fibred surface with fibres of genus g. Let $u_f$ $u_{f}$ be its unitary rank, i.e. the rank of the flat unitary part in the second Fujita decomposition. We study in detail the case when $u_f$ $u_{f}$ is maximal, i.e. $u_f=g-1$ $u_{f} = g - 1$ . In this case necessarily $g\le 6$ $g \leq 6$ , but examples in genus 5 and 6 are not known, and conjecturally do not exist. We prove a strong slope inequality for these extremal cases. We then use this inequality, together with results on trigonal curves, to give new constraints on the case $g=6$ $g = 6$ , $u_f=5$ $u_{f} = 5$ . In particular, we prove that the index of the surface is always strictly positive and give strong limitations on the possible classes of the relative canonical divisor.