GermanWikipedia - /page/entry.mu

<!DOCTYPE html>

<title>Divisor</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Divisor</span></h1>

</header>

</div>

<p>Der Begriff des <b>Divisors</b> spielt in der <a href="Algebraische_Geometrie" title="Algebraische Geometrie">algebraischen Geometrie</a> und der <a href="Funktionentheorie" title="Funktionentheorie">komplexen Analysis</a> eine wichtige Rolle bei der Untersuchung <a href="Algebraische_Variet%C3%A4t" title="Algebraische Varietät">algebraischer Varietäten</a> bzw. <a href="Komplexe_Mannigfaltigkeit" title="Komplexe Mannigfaltigkeit">komplexer Mannigfaltigkeiten</a> und der darauf definierten Funktionen. Unterschieden werden müssen dabei der <i>Weil-Divisor</i> und der <i>Cartier-Divisor</i>, welche in bestimmten Fällen übereinstimmen.

</p><p>Ursprünglich kommt dem Divisor im eindimensionalen Fall die Bedeutung zu, die Null- und Polstellenmenge einer <a href="Rationale_Funktion" title="Rationale Funktion">rationalen</a> bzw. <a href="Meromorphe_Funktion" title="Meromorphe Funktion">meromorphen</a> Funktion vorzuschreiben, und es stellt sich die Frage, für welche Divisoren eine solche Realisierung möglich ist, was eng mit der Geometrie der Varietät bzw. Mannigfaltigkeit verknüpft ist.

</p>

<div class="mw-heading mw-heading2"><h2 id="Eindimensionaler_Fall">Eindimensionaler Fall</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Funktionentheorie">Funktionentheorie</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Definition">Definition</h4></div>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega \subseteq \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd24e469a556af8609cf27aea06f0d2a885b35c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.455ex; height:2.343ex;" alt="{\displaystyle \Omega \subseteq \mathbb {C} }" loading="lazy"></span> ein <a href="Gebiet_(Mathematik)" title="Gebiet (Mathematik)">Gebiet</a> oder eine <a href="Riemannsche_Fl%C3%A4che" title="Riemannsche Fläche">riemannsche Fläche</a>. Eine Abbildung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\colon \Omega \rightarrow \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D\colon \Omega \rightarrow \mathbb {Z} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6be8ac44aac5b5da7c09da27dbebbecdec9d285f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.801ex; height:2.176ex;" alt="{\displaystyle D\colon \Omega \rightarrow \mathbb {Z} }" loading="lazy"></span> heißt Divisor in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span>, falls ihr <a href="Tr%C3%A4ger_(Mathematik)" title="Träger (Mathematik)">Träger</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:=\{z\in \Omega \,:\,D(z)\neq 0\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T:=\{z\in \Omega \,:\,D(z)\neq 0\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee1eb155545379bfd64f90e86205e80819efd205.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.107ex; height:2.843ex;" alt="{\displaystyle T:=\{z\in \Omega \,:\,D(z)\neq 0\}}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> <a href="Abgeschlossene_Menge" title="Abgeschlossene Menge">abgeschlossen</a> und <a href="Diskrete_Teilmenge" title="Diskrete Teilmenge">diskret</a> ist. Die Menge aller Divisoren auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> bildet bezüglich der Addition eine abelsche Gruppe, die mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Div} (\Omega )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Div} (\Omega )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b2e9ba334b4e439456d389ec2a0eb8485e720c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.137ex; height:2.843ex;" alt="{\displaystyle \operatorname {Div} (\Omega )}" loading="lazy"></span> bezeichnet wird. Auf dieser Gruppe führt man eine <a href="Partielle_Ordnung" class="mw-redirect" title="Partielle Ordnung">partielle Ordnung</a> ein. Seien <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D,D'\in \operatorname {Div} (\Omega )}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D,D'\in \operatorname {Div} (\Omega )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b45bcec693ac183f83fbccedcdd30fcb6e05476.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.545ex; height:3.009ex;" alt="{\displaystyle D,D'\in \operatorname {Div} (\Omega )}" loading="lazy"></span>, dann setzt man <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\leq D'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D\leq D'}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c9b1a3d800f578a837984e9649e1a1756029eea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.632ex; height:2.676ex;" alt="{\displaystyle D\leq D'}" loading="lazy"></span>, falls <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(x)\leq D'(x)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D(x)\leq D'(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a48bb4aa78b420ae44136c066e8899009a719b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.91ex; height:3.009ex;" alt="{\displaystyle D(x)\leq D'(x)}" loading="lazy"></span> für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \Omega }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40de8fa70c126001826d34da2a114349d8476bb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \Omega }" loading="lazy"></span> gilt.

</p>

<div class="mw-heading mw-heading4"><h4 id="Hauptdivisor">Hauptdivisor</h4></div>

<p>Zu jeder von Null verschiedenen <a href="Meromorphe_Funktion" title="Meromorphe Funktion">meromorphen</a> Funktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \Omega \to \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon \Omega \to \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aecb41fd1a98cfb0e2dd9bf276e0ae1da00db6f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.283ex; height:2.509ex;" alt="{\displaystyle f\colon \Omega \to \mathbb {C} }" loading="lazy"></span> kann ein Divisor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8bea8bbec80317513538b12e2ec1ad420b9e1bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.088ex; height:2.843ex;" alt="{\displaystyle (f)}" loading="lazy"></span> definiert werden, indem der Divisor jedem Punkt aus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> die Null- beziehungsweise die Polstellenordnung zuordnet:

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(f\right)(x):={\begin{cases}0,&f{\mbox{ holomorph und nicht Null in }}x\\k,&f{\mbox{ hat eine Nullstelle von Ordnung }}k{\mbox{ in }}x\\-k,&f{\mbox{ hat eine Polstelle von Ordnung }}k{\mbox{ in }}x\end{cases}}}">

<mrow>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext> holomorph und nicht Null in </mtext>

</mstyle>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext> hat eine Nullstelle von Ordnung </mtext>

</mstyle>

</mrow>

</mstyle>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext> hat eine Polstelle von Ordnung </mtext>

</mstyle>

</mrow>

</mstyle>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(f\right)(x):={\begin{cases}0,&f{\mbox{ holomorph und nicht Null in }}x\\k,&f{\mbox{ hat eine Nullstelle von Ordnung }}k{\mbox{ in }}x\\-k,&f{\mbox{ hat eine Polstelle von Ordnung }}k{\mbox{ in }}x\end{cases}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85c0a8c2293321aad218fb219bf2c070f97db319.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:59.177ex; height:8.509ex;" alt="{\displaystyle \left(f\right)(x):={\begin{cases}0,&f{\mbox{ holomorph und nicht Null in }}x\\k,&f{\mbox{ hat eine Nullstelle von Ordnung }}k{\mbox{ in }}x\\-k,&f{\mbox{ hat eine Polstelle von Ordnung }}k{\mbox{ in }}x\end{cases}}}" loading="lazy"></span>

</p><p>Ein Divisor, der gleich dem Divisor einer meromorphen Funktion ist, heißt <i>Hauptdivisor</i>.

</p><p>Der <a href="Weierstra%C3%9Fscher_Produktsatz" title="Weierstraßscher Produktsatz">weierstraßsche Produktsatz</a> besagt, dass in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"></span> jeder Divisor ein Hauptdivisor ist. In einer kompakten, riemannschen Fläche gilt dies jedoch nicht mehr und ist vom Geschlecht der Fläche abhängig. Dies wird im Artikel <a href="Satz_von_Riemann-Roch" title="Satz von Riemann-Roch">Satz von Riemann-Roch</a> näher erläutert.

</p>

<div class="mw-heading mw-heading3"><h3 id="Algebraische_Kurven">Algebraische Kurven</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> eine ebene <a href="Algebraische_Kurve" title="Algebraische Kurve">algebraische Kurve</a>. Eine formale Summe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{P\in C}a_{P}\cdot P,\;a_{P}\in \mathbb {Z} }">

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{P\in C}a_{P}\cdot P,\;a_{P}\in \mathbb {Z} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1729c67b09bac572bb5a7ea6b8de79f4d03dc37f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.54ex; height:3.009ex;" alt="{\displaystyle \textstyle \sum _{P\in C}a_{P}\cdot P,\;a_{P}\in \mathbb {Z} }" loading="lazy"></span> heißt Divisor in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span>, falls <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{P}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{P}=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fa4f1fc9a71f05f94e94489443f762d4feccb04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.957ex; height:2.509ex;" alt="{\displaystyle a_{P}=0}" loading="lazy"></span> außer für endlich viele <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span>. Durch punktweise Addition wird die Menge aller Divisoren in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> zu einer <a href="Freie_Gruppe" title="Freie Gruppe">freien</a> <a href="Abelsche_Gruppe" title="Abelsche Gruppe">abelschen Gruppe</a>.

</p><p>Analog zur o. g. Definition definiert man für eine rationale Funktion den Divisor der Funktion. Ein Divisor, der gleich dem Divisor einer rationalen Funktion ist, heißt <i>Hauptdivisor</i>.

</p><p>Im Falle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=\mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C=\mathbb {C} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a860e6c6cbc56a4b82480ba898bd364e081da30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.543ex; height:2.176ex;" alt="{\displaystyle C=\mathbb {C} }" loading="lazy"></span> ist für einen Divisor die Abbildung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\mapsto a_{P}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\mapsto a_{P}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f091a2b39ba61512ef0f9e02b98a10a74b66f8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.056ex; height:2.509ex;" alt="{\displaystyle P\mapsto a_{P}}" loading="lazy"></span> ein Divisor im Sinne der Funktionentheorie. Allerdings gibt es Divisoren im Sinne der Funktionentheorie, die nicht auf diese Weise entstehen, da dort <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{P}\neq 0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{P}\neq 0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6354e4e0fc32e062ad4d86fd6c335962b1d1f7ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.957ex; height:2.676ex;" alt="{\displaystyle a_{P}\neq 0}" loading="lazy"></span> für unendlich viele <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> (die allerdings keinen <a href="H%C3%A4ufungspunkt" title="Häufungspunkt">Häufungspunkt</a> haben dürfen) zugelassen ist.

</p>

<div class="mw-heading mw-heading2"><h2 id="Allgemeine_Definition">Allgemeine Definition</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Weil-Divisor">Weil-Divisor</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> ein noethersches integres separiertes <a href="Schema_(algebraische_Geometrie)" title="Schema (algebraische Geometrie)">Schema</a>, regulär in <a href="Kodimension" title="Kodimension">Kodimension</a> 1. Ein <i>Primdivisor</i> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> ist ein abgeschlossenes ganzes Unter-Schema der Kodimension Eins. Ein <i>Weil-Divisor</i> (nach <a href="Andr%C3%A9_Weil" title="André Weil">André Weil</a>) ist dann ein Element der frei erzeugten abelschen Gruppe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Div} \,X}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Div} \,X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c995357f171e1784a6d353eefead8c3a60c4c7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.017ex; height:2.176ex;" alt="{\displaystyle \mathrm {Div} \,X}" loading="lazy"></span> der Primdivisoren und wird meistens als formale Summe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle D=\sum _{j}a_{j}\cdot Y_{j},\;a_{j}\in \mathbb {Z} ,}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle D=\sum _{j}a_{j}\cdot Y_{j},\;a_{j}\in \mathbb {Z} ,}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0bf4b2bb14ef73f1b7ad0348209c583a0960beb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:23.709ex; height:3.343ex;" alt="{\displaystyle \textstyle D=\sum _{j}a_{j}\cdot Y_{j},\;a_{j}\in \mathbb {Z} ,}" loading="lazy"></span> geschrieben, wobei nur endlich viele <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{j}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0096fb78d6843c9fb67a840dc796b61ad93eec2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.14ex; height:2.343ex;" alt="{\displaystyle a_{j}}" loading="lazy"></span> von Null verschieden sind.

</p>

<ul><li>Ein Weil-Divisor heißt <i>effektiv</i> (oder <i>positiv</i>), wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{j}\geq 0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{j}\geq 0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa5df7b71736203ed48927284e85ce627066fe8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.4ex; height:2.843ex;" alt="{\displaystyle a_{j}\geq 0}" loading="lazy"></span> für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span> gilt.</li></ul>

<ul><li>Ein Weil-Divisor heißt <i>Hauptdivisor</i>, falls er gleich dem Divisor einer von Null verschiedenen rationalen Funktion ist: Sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> eine rationale Funktion auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>, von Null verschieden. Für jeden Primdivisor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> bezeichne <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(f,Y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(f,Y)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e590d365260a160091b001edc1d5edc7c3a73b0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\displaystyle v(f,Y)}" loading="lazy"></span> die Bewertung von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> im <a href="Diskreter_Bewertungsring" title="Diskreter Bewertungsring">diskreten Bewertungsring</a>, der zu einem generischen Punkt von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> gehört. Die Bewertung ist von der Wahl des generischen Punktes unabhängig. Im eindimensionalen Fall entspricht die Bewertung dem Grad der Null- bzw. <a href="Polstelle" title="Polstelle">Polstelle</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> in diesem Punkt. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle (f):=\sum _{Y}v(f,Y)\cdot Y}">

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle (f):=\sum _{Y}v(f,Y)\cdot Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/844d9afa43c201f7244a68f0218a82cf50ca6ddc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.635ex; height:2.843ex;" alt="{\displaystyle \textstyle (f):=\sum _{Y}v(f,Y)\cdot Y}" loading="lazy"></span> heißt dann <i>Divisor von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span></i> und definiert tatsächlich einen Weil-Divisor, die Summanden sind nur für endlich viele Primdivisoren von Null verschieden.</li></ul>

<ul><li>Zwei Weil-Divisoren heißen <i>linear äquivalent</i>, falls ihre Differenz ein Hauptdivisor ist. Der Quotient von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Div} \,X}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Div} \,X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c995357f171e1784a6d353eefead8c3a60c4c7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.017ex; height:2.176ex;" alt="{\displaystyle \mathrm {Div} \,X}" loading="lazy"></span> bezüglich dieser Äquivalenz ist die <i>Divisorenklassengruppe</i> und wird mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Cl} \,X}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Cl} \,X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9b5e7c479ece5c39a04c3eb519b788d385d32ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.692ex; height:2.176ex;" alt="{\displaystyle \mathrm {Cl} \,X}" loading="lazy"></span> bezeichnet.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Cartier-Divisor">Cartier-Divisor</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> eine <a href="Komplexe_Mannigfaltigkeit" title="Komplexe Mannigfaltigkeit">komplexe Mannigfaltigkeit</a> bzw. eine <a href="Algebraische_Variet%C3%A4t" title="Algebraische Varietät">algebraische Varietät</a> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6ae2ed4058fb748a183d9ada8aea50a00d0c89f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.85ex; height:2.176ex;" alt="{\displaystyle {\mathcal {O}}}" loading="lazy"></span> bezeichne die <a href="Garbe_(Mathematik)" title="Garbe (Mathematik)">Garbe</a> der holomorphen bzw. algebraischen Funktionen auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2cc2abebd45ec020509a0ec548b67c9a2cb7cecd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\mathcal {M}}}" loading="lazy"></span> bezeichne die Garbe der meromorphen bzw. rationalen Funktionen auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>. Die Quotienten-Garbe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}:={\mathcal {M}}^{*}/{\mathcal {O}}^{*}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}:={\mathcal {M}}^{*}/{\mathcal {O}}^{*}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14b4ca29d503d76d611540c6ce07a609d557543b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.449ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}:={\mathcal {M}}^{*}/{\mathcal {O}}^{*}}" loading="lazy"></span> heißt <i>Garbe der Divisoren</i>, und ein <a href="Faserb%C3%BCndel" title="Faserbündel">Schnitt</a> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3277962e1959c3241fb1b70c7f0ac6dcefebd966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}" loading="lazy"></span> heißt <b>Cartier-Divisor</b> (nach <a href="Pierre_Cartier_(Mathematiker)" title="Pierre Cartier (Mathematiker)">Pierre Cartier</a>), meist nur als <i>Divisor</i> bezeichnet. Die Menge aller Schnitte <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (X,{\mathcal {M}}^{*}/{\mathcal {O}}^{*})}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (X,{\mathcal {M}}^{*}/{\mathcal {O}}^{*})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35722b6c634a24cc4baba887592b8f3e4c1c864e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.187ex; height:2.843ex;" alt="{\displaystyle \Gamma (X,{\mathcal {M}}^{*}/{\mathcal {O}}^{*})}" loading="lazy"></span> bildet eine abelsche Gruppe.

</p>

<ul><li>Ein Cartier-Divisor heißt <i>Hauptdivisor</i>, falls er im Bild der natürlichen Abbildung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (X,{\mathcal {M}}^{*})\to \Gamma (X,{\mathcal {M}}^{*}/{\mathcal {O}}^{*})}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (X,{\mathcal {M}}^{*})\to \Gamma (X,{\mathcal {M}}^{*}/{\mathcal {O}}^{*})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6d1d9354fc28a9af083359244462d49a2b24ce0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.922ex; height:2.843ex;" alt="{\displaystyle \Gamma (X,{\mathcal {M}}^{*})\to \Gamma (X,{\mathcal {M}}^{*}/{\mathcal {O}}^{*})}" loading="lazy"></span> liegt, also der Divisor einer nicht-verschwindenden meromorphen Funktion ist.</li></ul>

<ul><li>Zwei Cartier-Divisoren heißen <i>linear äquivalent</i>, falls ihr Quotient ein Hauptdivisor ist. Der Quotient bezüglich dieser Äquivalenz wird mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {CaCl} \,X}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {CaCl} \,X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e1271cd5ade273516bfe522fbf2d9f991cecfa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.533ex; height:2.176ex;" alt="{\displaystyle \mathrm {CaCl} \,X}" loading="lazy"></span> bezeichnet.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Beziehung_zwischen_Cartier-_und_Weil-Divisoren">Beziehung zwischen Cartier- und Weil-Divisoren</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> ein noethersches integres separiertes Schema, dessen <a href="Lokaler_Ring" title="Lokaler Ring">lokale Ringe</a> alle <a href="Faktorieller_Ring" title="Faktorieller Ring">faktoriell</a> sind. Dann ist die Gruppe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Div} \,X}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Div} \,X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c995357f171e1784a6d353eefead8c3a60c4c7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.017ex; height:2.176ex;" alt="{\displaystyle \mathrm {Div} \,X}" loading="lazy"></span> der Weil-Divisoren auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> isomorph zur Gruppe der Cartier-Divisoren <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (X,{\mathcal {M}}^{*}/{\mathcal {O}}^{*})}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (X,{\mathcal {M}}^{*}/{\mathcal {O}}^{*})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35722b6c634a24cc4baba887592b8f3e4c1c864e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.187ex; height:2.843ex;" alt="{\displaystyle \Gamma (X,{\mathcal {M}}^{*}/{\mathcal {O}}^{*})}" loading="lazy"></span>. Dieser Isomorphismus erhält die Eigenschaft, Hauptdivisor zu sein und führt die Quotientengruppen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Cl} \,X}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Cl} \,X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9b5e7c479ece5c39a04c3eb519b788d385d32ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.692ex; height:2.176ex;" alt="{\displaystyle \mathrm {Cl} \,X}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {CaCl} \,X}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {CaCl} \,X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e1271cd5ade273516bfe522fbf2d9f991cecfa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.533ex; height:2.176ex;" alt="{\displaystyle \mathrm {CaCl} \,X}" loading="lazy"></span> ineinander über.

</p>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<div class="sisterproject" style="margin:0.1em 0 0 0;"><span class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Wiktionary"></span></span></span><b><a href="https://de.wiktionary.org/wiki/Divisor" class="extiw external" title="wikt:Divisor">Wiktionary: Divisor</a></b> – Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen</div>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Joseph L. Taylor: <i>Several Complex Variables with Connections to Algebraic Geometry and Lie Groups.</i> American Mathematical Society 2002, ISBN 0-8218-3178-X</li>

<li>William Fulton: <i>Algebraic Curves. An Introduction to Algebraic Geometry.</i> Mathematics lecture note series, 30. Benjamin/Cummings, New York 1969, ISBN 0-201-51010-3</li>

<li>Robin Hartshorne: <i>Algebraic Geometry</i>. Springer-Verlag 1977. ISBN 0-387-90244-9</li>

<li>Reinhold Remmert, Georg Schumacher: <i>Funktionentheorie 2.</i> Springer, Berlin 2007, ISBN 978-3-540-57052-3</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2024-10-14" href="https://de.wikipedia.org/wiki/?title=Divisor&oldid=249410754">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>