alarraya/Curvas
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Añadir bibliografía

Browse Source
This commit is contained in:
Miguel Marco 2026-05-25 11:48:39 +02:00
parent a2f554435c
commit a2d54b51d3
Signed by: mmarco
GPG Key ID: DD0114EDB7410592
1 changed files with 73 additions and 50 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

29
generaldegree.tex
View File

@ -33,14 +33,15 @@ decorations.markings, calc}
\maketitle \maketitle
\begin{abstract} \begin{abstract}
We give a explicit construction of a $2$- dimensional $\Delta$-complex that is a strong deformation retract of the complement of a Fermat curve of degree $d$ in the complex projective space.
Due to a result by Androtti and Frankel \cite{andreotti19}, we can see that the complement o a complex projective curve has the homotopy type of a $2$-dimensional CW complex. However, no general method has been given to compute explicitly this complex. Here we give a explicit construction of a $2$- dimensional $\Delta$-complex that is a strong deformation retract of the complement of a Fermat curve of degree $d$ in the complex projective space.
The retraction is performed in several steps, using the branched cover structure of the Fermat curves to the degree $1$ case. The retraction is performed in several steps, using the branched cover structure of the Fermat curves to the degree $1$ case.
\end{abstract} \end{abstract}
\section{Definition of the problem} \section{Definition of the problem}
Given $C$ a proyective curve in $\CC\PP^2$, we know that the complement of the curve has the homotopy type of a CW-complex of real dimension $2$, our problem is how to find an effective model for said CW-complex. Given $C$ a proyective curve in $\CC^2$, Libgober gave in \cite{libgober1986} a construction of a $2-$ dimensional CW-complex that is a deformation retract of its complement. However, it is not clear how to extend that construction to the projective case (that is a curve in $\CC\PP^2$).
In particular, we will focus on the case of smooth curves. In this work, we give a different construction to produce a specific $\Delta$-complex that is a deformation retract of the complement of a Fermat curve (which is equivalent to the case of general smooth curves).
\section{Smooth curves} \section{Smooth curves}
Given a smooth curve of degree $d$ in $\PP^2$, we know it can be moved to a curve of the form Given a smooth curve of degree $d$ in $\PP^2$, we know it can be moved to a curve of the form
\begin{equation} \begin{equation}
@ -685,4 +686,26 @@ Lastly, we will have $4$ $2-$simplices, $X_{[1:1:1]}=X_{[-1:-1:-1]}$, $X_{[1:1:-
The face maps will be obtained by passing to the projective the face maps of $\Delta^d$. The face maps will be obtained by passing to the projective the face maps of $\Delta^d$.
\end{example} \end{example}
\begin{thebibliography}{99}
\bibitem{andreotti1959}
A. Andreotti and T. Frankel,
\textit{The Lefschetz theorem on hyperplane sections},
Annals of Mathematics,
Vol. 69, No. 3,
pp. 713--717,
1959.
\bibitem{libgober1986}
A. Libgober,
\textit{On the homotopy type of the complement to plane algebraic curves},
J. Reine Angew. Math.,
Vol. 367,
pp. 103--114,
1986.
DOI: 10.1515/crll.1986.367.103
\end{thebibliography}
\end{document} \end{document}