Let S be a smooth affine surface of logarithmic Kodaira dimension one and let (V, D) be a pair of a smooth projective surface V and a simple normal crossing divisor D on V such that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_984_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="119" /> </InlineMediaObject> <EquationSource Format="TEX">$V \setminus \operatorname {Supp}D = S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mo>Supp</mo> <mi>D</mi> <mo>=</mo> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we consider the logarithmic multicanonical system <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_984_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">$|m(K_V + D)|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>m</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mi>V</mi> </msub> <mo>+</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>. We prove that, for any <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_984_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$m \ge 8$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>8</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_984_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">$|m(K_V+D)|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>m</mi> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mi>V</mi> </msub> <mo>+</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation> gives an <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_984_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <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>-fibration from V onto a smooth projective curve.

Logarithmic multicanonical systems of smooth affine surfaces of logarithmic Kodaira dimension one

期刊论文
发表日期：

2025年2月13日

查看全文

Hideo Kojima

摘要

Let S be a smooth affine surface of logarithmic Kodaira dimension one and let (V, D) be a pair of a smooth projective surface V and a simple normal crossing divisor D on V such that $V \setminus \operatorname {Supp}D = S$ $V \ Supp D = S$ . In this paper, we consider the logarithmic multicanonical system $|m(K_V + D)|$ $| m (K_{V} + D) |$ . We prove that, for any $m \ge 8$ $m \geq 8$ , $|m(K_V+D)|$ $| m (K_{V} + D) |$ gives an $\mathbb {P}^1$ $P^{1}$ -fibration from V onto a smooth projective curve.