<p>We complete the Kodaira classification of the moduli spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\overline{\mathcal {M}}_{g,n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover> <mi mathvariant="script">M</mi> <mo>¯</mo> </mover> <mrow> <mi>g</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> of curves with marked points in genus <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$g=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, by proving that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\overline{\mathcal {M}}_{3,n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover> <mi mathvariant="script">M</mi> <mo>¯</mo> </mover> <mrow> <mn>3</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is of general type for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n \ge 15$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>15</mn> </mrow> </math></EquationSource> </InlineEquation>. We prove that the singularities of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\overline{\mathcal {M}}_{3,n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover> <mi mathvariant="script">M</mi> <mo>¯</mo> </mover> <mrow> <mn>3</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> impose no adjunction conditions for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and that the canonical class of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\overline{\mathcal {M}}_{3,n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover> <mi mathvariant="script">M</mi> <mo>¯</mo> </mover> <mrow> <mn>3</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is big for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n \ge 15$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>15</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

The Kodaira classification of the moduli space of pointed curves in genus 3

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发表日期：

2026年5月11日

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Ruben de Preter

摘要

We complete the Kodaira classification of the moduli spaces $\overline{\mathcal {M}}_{g,n}$ ${\bar{M}}_{g, n}$ of curves with marked points in genus $g=3$ $g = 3$ , by proving that $\overline{\mathcal {M}}_{3,n}$ ${\bar{M}}_{3, n}$ is of general type for $n \ge 15$ $n \geq 15$ . We prove that the singularities of $\overline{\mathcal {M}}_{3,n}$ ${\bar{M}}_{3, n}$ impose no adjunction conditions for $n \ge 1$ $n \geq 1$ and that the canonical class of $\overline{\mathcal {M}}_{3,n}$ ${\bar{M}}_{3, n}$ is big for $n \ge 15$ $n \geq 15$ .