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<p>First we construct a cubic 4-fold whose singularities are 11 cusps and which has an action of the Mathieu group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(M_{11}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mn>11</mn> </msub> </math></EquationSource> </InlineEquation>, all over the ternary field <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {F}_3\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>. We next consider a certain moduli space of bundles on a supersingular K3 surface of Artin invariant one in characteristic 3. We show that it has 275 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((-\,2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mspace width="0.166667em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> Mukai vectors which form the McLaughlin graph, and ask questions on it and on its relation with our <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(M_{11}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mn>11</mn> </msub> </math></EquationSource> </InlineEquation>-cubic 4-fold.</p>

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Cubic fourfolds with eleven cusps and a related moduli space

  • 期刊论文

  • 发表日期：

    2026年6月19日

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  • Shigeru Mukai

摘要

First we construct a cubic 4-fold whose singularities are 11 cusps and which has an action of the Mathieu group \(M_{11}\) M 11 , all over the ternary field \(\mathbb {F}_3\) F 3 . We next consider a certain moduli space of bundles on a supersingular K3 surface of Artin invariant one in characteristic 3. We show that it has 275 \((-\,2)\) ( - 2 ) Mukai vectors which form the McLaughlin graph, and ask questions on it and on its relation with our \(M_{11}\) M 11 -cubic 4-fold.