LaTeX Forum

\documentclass{scrartcl}
\usepackage[english]{babel}

\usepackage{graphicx}
\usepackage{amsmath, amsfonts, amsthm, geometry}

\usepackage[utf8]{inputenc} % Braucht es schon seit 2018 nicht mehr!
\usepackage{fancyhdr, lastpage} % Wird im Beispiel gar nicht verwendet, gehört also eigentlich auch nicht rein.

\usepackage{wrapfig2, framed, caption}

\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\PP}{\mathbb{P}}

\begin{document}

\section{The Shadow of a Cubic}
\begin{wrapfigure}[14]{r}{0.4\linewidth} 

     \begin{framed}\raggedleft%\centering
     
     \rule{5cm}{4cm}
       \caption{Fermat cubic}
       \label{3linesferm}  
\end{framed}

\end{wrapfigure}
It is a historical and remarkable fact, that any smooth Variety of degree $3$ in $\mathbb{P}^3$ contains exactly 27 lines. We call those Varieties Cubics and a line is a subvariety defined by two homogeneous polynomial of degree 1
\section{Theomrem}
There is exactly 27 lines on a smooth Cubic in $\mathbb{P}^3$.

\subsection{Proof}
To show this, we will use, that the fermat cubic $V(x_0^3+x_1^3+x_2^3+x_3^3)$ has exactly 27 lines. Afterwards we will show, that all smooth cubics have the same amount of lines. We will do that by showing, that 
\begin{equation*}
       \mathcal{M}=\{(X,L)|X \text{ is a smooth cubic}, \} ,
   \end{equation*} is a zarisky closed subset of... 

The fermat cubic has 27 Lines.

\end{document}

\documentclass{article}
\usepackage[english]{babel}

\usepackage{graphicx}
\usepackage{amsmath, amsfonts, amsthm, geometry}

\usepackage[utf8]{inputenc} % Braucht es schon seit 2018 nicht mehr!
\usepackage{fancyhdr, lastpage} % Wird im Beispiel gar nicht verwendet, gehört also eigentlich auch nicht rein.

\usepackage{wrapfig2, framed, caption}

\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\PP}{\mathbb{P}}

\begin{document}

\section{The Shadow of a Cubic}
\begin{wrapfigure}[14]{r}{0.4\linewidth} 

     \begin{framed}\raggedleft%\centering
     
     \rule{5cm}{4cm}
       \caption{Fermat cubic}
       \label{3linesferm}  
\end{framed}

\end{wrapfigure}
It is a historical and remarkable fact, that any smooth Variety of degree $3$ in $\mathbb{P}^3$ contains exactly 27 lines. We call those Varieties Cubics and a line is a subvariety defined by two homogeneous polynomial of degree 1
\section{Theomrem}
There is exactly 27 lines on a smooth Cubic in $\mathbb{P}^3$.

\subsection{Proof}
To show this, we will use, that the fermat cubic $V(x_0^3+x_1^3+x_2^3+x_3^3)$ has exactly 27 lines. Afterwards we will show, that all smooth cubics have the same amount of lines. We will do that by showing, that 
\begin{equation*}
       \mathcal{M}=\{(X,L)|X \text{ is a smooth cubic}, \} ,
   \end{equation*} is a zarisky closed subset of... 

The fermat cubic has 27 Lines.

\end{document}

\documentclass{amsart}
\usepackage[english]{babel}




\usepackage{graphicx}
\usepackage{amsmath, amsfonts, amsthm, geometry}

\usepackage[utf8]{inputenc} %kann sonderzeichen aus dem Text lesen
\usepackage{fancyhdr, lastpage} %Kopfzeile und referenz zur letzten Seite

\usepackage{wrapfig, framed, caption}


\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\PP}{\mathbb{P}}

\begin{document}

\section{The Shadow of a Cubic}
\begin{wrapfigure}[14]{r}{0.4\linewidth} 

     \begin{framed}\raggedleft%\centering
     
     \rule{5cm}{4cm}
       \caption{Fermat cubic}
       \label{3linesferm}  
\end{framed}

\end{wrapfigure}
It is a historical and remarkable fact, that any smooth Variety of degree $3$ in $\mathbb{P}^3$ contains exactly 27 lines. We call those Varieties Cubics and a line is a subvariety defined by two homogeneous polynomial of degree 1
\section{Theomrem}
There is exactly 27 lines on a smooth Cubic in $\mathbb{P}^3$.


\subsection{Proof}
To show this, we will use, that the fermat cubic $V(x_0^3+x_1^3+x_2^3+x_3^3)$ has exactly 27 lines. Afterwards we will show, that all smooth cubics have the same amount of lines. We will do that by showing, that 
\begin{equation*}
       \mathcal{M}=\{(X,L)|X \text{ is a smooth cubic}, \} ,
   \end{equation*} is a zarisky closed subset of... 




The fermat cubic has 27 Lines.


\end{document}

\documentclass{amsart}
\usepackage[english]{babel}




\usepackage{graphicx}
\usepackage{amsmath, amsfonts, amsthm, geometry}

\usepackage[utf8]{inputenc} %kann sonderzeichen aus dem Text lesen
\usepackage{fancyhdr, lastpage} %Kopfzeile und referenz zur letzten Seite

\usepackage{wrapfig, framed, caption}


\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\PP}{\mathbb{P}}

\begin{document}

\section{The Shadow of a Cubic}
\begin{wrapfigure}[14]{r}{0.4\linewidth} 

     \begin{framed}\raggedleft%\centering
     
     \rule{5cm}{4cm}
       \caption{Fermat cubic}
       \label{3linesferm}  
\end{framed}

\end{wrapfigure}
It is a historical and remarkable fact, that any smooth Variety of degree $3$ in $\mathbb{P}^3$ contains exactly 27 lines. We call those Varieties Cubics and a line is a subvariety defined by two homogeneous polynomial of degree 1
\section{Theomrem}
There is exactly 27 lines on a smooth Cubic in $\mathbb{P}^3$.


\subsection{Proof}
To show this, we will use, that the fermat cubic $V(x_0^3+x_1^3+x_2^3+x_3^3)$ has exactly 27 lines. Afterwards we will show, that all smooth cubics have the same amount of lines. We will do that by showing, that 
\begin{equation*}
       \mathcal{M}=\{(X,L)|X \text{ is a smooth cubic}, L \text{ is a line in X} \} \subset U \times G(2,4),
   \end{equation*} is a zarisky closed subset of... 




The fermat cubic has 27 Lines.


\end{document}