intro

biblio.bib

@@ -0,0 +1,47 @@
+@article {acgm,
+    AUTHOR = {Artal, E. and Cogolludo, J.I. and {Guerville-Ball\'e}, B. and Marco, M.\'A.},
+     TITLE = {An arithmetic {Z}ariski pair of line arrangements with
+              non-isomorphic fundamental group},
+   JOURNAL = {Rev. R. Acad. Cienc. Exactas F\'is. Nat. Ser. A Mat. RACSAM},
+  FJOURNAL = {Revista de la Real Academia de Ciencias Exactas, F\'isicas y
+              Naturales. Serie A. Matematicas. RACSAM},
+    VOLUME = {111},
+      YEAR = {2017},
+    NUMBER = {2},
+     PAGES = {377--402},
+      ISSN = {1578-7303,1579-1505},
+       DOI = {10.1007/s13398-016-0298-y},
+       URL = {https://doi.org/10.1007/s13398-016-0298-y},
+}
+
+@Article{Libgober1986,
+  author   = {A. Libgober},
+  journal  = {J. Reine Angew. Math.},
+  title    = {On the homotopy type of the complement to plane algebraic curves},
+  year     = {1986},
+  pages    = {103--114},
+  volume   = {367},
+  fjournal = {Journal f{\"u}r die Reine und Angewandte Mathematik},
+}
+
+@article {andreotti1959,
+    AUTHOR = {Andreotti, A. and Frankel, T.},
+     TITLE = {The {L}efschetz theorem on hyperplane sections},
+   JOURNAL = {Ann. of Math. (2)},
+  FJOURNAL = {Annals of Mathematics. Second Series},
+    VOLUME = {69},
+      YEAR = {1959},
+     PAGES = {713--717},
+      ISSN = {0003-486X},
+       DOI = {10.2307/1970034},
+       URL = {https://doi.org/10.2307/1970034},
+}
+
+@Manual{Sagemath,
+  title        = {Sage {M}athematics {S}oftware ({V}ersion 10.10)},
+  author       = {Stein, W.A. and others},
+  note         = {{\tt http://www.sagemath.org}},
+  organization = {The Sage Development Team},
+  year         = {2026},
+  key          = {Sage},
+}

generaldegreeHypersurfaces.tex

@@ -35,7 +35,7 @@ decorations.markings, calc}
 \numberwithin{equation}{section}
 
 %opening
-\title[Retraction of the complement of a smooth curve]{Retraction of the complement of smooth projective hypersurfaces to a $n$-dimensional $\Delta$-complex.}
+\title[Retraction of the complement of a smooth curve]{Retraction of the complement of smooth projective hypersurfaces to an $n$-dimensional $\Delta$-complex.}
 \author[E. Artal]{Enrique Artal Bartolo} 
 \address{Departamento de Matem\'{a}ticas, IUMA \\
 Universidad de Zaragoza \\
@@ -71,17 +71,29 @@ MICIU/AEI/10.13039/501100011033 and CEX2021-001142-S-20-9.}
 
 
 \begin{abstract}
+We give an explicit strong deformation retraction of the complement of a Fermat hypersurface of degree $d$ in the complex projective space of dimension $n$ into a $\Delta$-complex of real dimension~$n$.
+%Due to a result by Andreotti and Frankel \cite{andreotti1959}, it can be seen that the complement of 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 $n$- dimensional $\Delta$-complex that is a strong deformation retract of the complement of a Fermat hypersurface of degree $d$ in the complex projective space of dimension $n$.
 
-Due to a result by Andreotti and Frankel \cite{andreotti1959}, it can be seen that the complement of 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 $n$- dimensional $\Delta$-complex that is a strong deformation retract of the complement of a Fermat hypersurface of degree $d$ in the complex projective space of dimension $n$.
+This retraction is performed in several steps, using the branched cover structure of the Fermat hypersurfaces to the degree $1$ case. Moreover, an implementation of the method in \texttt{Sagemath} is provided.
-
-The retraction is performed in several steps, using the branched cover structure of the Fermat hypersurfaces to the degree $1$ case.
 \end{abstract}
 
 \maketitle
-\section{Definition of the problem}
+\section*{Introduction}
-Given $C$ an affine hypersurface in $\CC^2$ (i.e. a curve), 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 $\CC\PP^2$).
+A result of Andreotti and Frankel \cite{andreotti1959} asserts that any affine smooth variety of complex dimension~$n$ has the homotopy type of a CW-complex of real dimension~$n$. Using the Segre embedding, it is not hard
+to check that the complement of a hypersurface in $\PP^n$ is isomorphic to an affine variety. 
+However, no general method has been given to compute explicitly this complex.
+Since there are hypersurfaces with equations in a number field $K$ whose embedding depend on the embedding $K\hookrightarrow\CC$ (even up to homotopy)~\cite{acgm}, a purely algebraic method cannot exist.
 
-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 hypersurface in $\CC\PP^n$. Since any smooth hypersurface is isotopic to a Fermat hypersurface, this particular case solves the problemfor a general smooth hypersurface.
+Possibly the only known method applies to the following particular case.
+%One main problem is to provide such a CW-complex.                     
+Given $C$ an affine 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, using braid monodromy which is not an algebraic construction. In the previous language a complement of an affine curve ins $\CC^2$ is the complement of a projective curve~$C$ in $\PP^2$, where $C$ contains a line $L$ (which can be seen as the line at infinity). 
+
+However, it is not clear how to extend that construction to the projective case, that is a curve in $\CC\PP^2$ such that no irreducible component is a line, e.g. a smooth curve of degree $>1$.
+
+In this work, we give a different construction to produce a specific $\Delta$-complex that is a deformation retract of the complement of a $d$-Fermat hypersurface in $\CC\PP^n$. Since any smooth hypersurface is isotopic to a Fermat hypersurface, this particular case gives a construction for the more general of smooth hypersurfaces.
+
+The actual construction starts with the trivial case $d=1$. In this case the complement is contractible and a point would suffice as $\Delta$-complex. Actually we will take a simplex of real dimension~$n$ which is also contractible.
+The main point is to understand the lifting of this complex by the Kummer cover $[x_0:\dots:x_n]\mapsto[x_0^d:\dots:x_n^d]$.
 
 
 \section{Smooth hypersurfaces}
@@ -145,21 +157,11 @@ and the restriction of $r_{1t}$ to $M_1\cap\RR^{n+1}$ is the inclusion.
 %where $\Re$ and $\Im$ take the real and imaginary part of each coordinate of a point, respectively.
 
 Note that for $(x_0,\dots,x_n)\in M_1$, we already have that
-<<<<<<< HEAD
-\begin{equation*}
-\begin{array}{lll}
-	\sum x^r_j & = & 1 \\
-	\sum x^i_j & = & 0 \\
-\end{array}
-\end{equation*}
-and therefore $r_1(x_0,\dots,x_n,t)\in M_1$ for all $t\in[0,1]$, and in particular, we get that $r_1(x_0,\dots,x_n,1)=(x^r_0,\dots,x^r_n)\in M_1\cap\RR^{n+1}$ for all $(x_0,\dots,x_n)\in M_1$. 
-=======
 \begin{align*}
 	\sum_{j=0}^n x^r_j & =  1 \\
 	\sum_{j=0}^n  x^i_j & =  0 \\
 \end{align*}
 and therefore $r_1(x_0,\dots,x_n,t)\in M_1$ for all $t\in[0,1]$, and in particular, we get that $r_1(x_0,\dots,x_n,1)=(x^r_0,\dots,x^r_n)\in M_1\cap\RR^{n+1}$ for all $(x_0,\dots,x_n)\in M_1$.
->>>>>>> 2a87fee69dfb1e6fdd6e608f9b4ee957cc3f5a74
 Let us summarize it.
 
 
@@ -447,23 +449,6 @@ We have then proved that $S_d$ is a strong deformation retract of $M_d$.
 	If $[x^d_0: \dots: x^d_n]$ is in $(+,\dots,+)$, the retraction is the identity, and therefore both representatives will stay the same and therefore go to the same class.
 	
 	For the rest of the  regions, we have
-<<<<<<< HEAD
-\begin{equation}
-\begin{array}{lll}
-\tilde{r}_2([x_0\xi_0:\dots:x_n\xi_n], t)& =& \left[\sqrt[d]{(1-t)x^d_0+t\frac{x^d_0\chi_0}{1-\sum x^d_j(1-\chi_j)}}\xi_0:\dots:\sqrt[d]{(1-t)x^d_n+t\frac{x^d_n\chi_n}{1-\sum x^d_j(1-\chi_j)}}\xi_n\right]
-\end{array}
-\end{equation}
-
-
-Then, for the point $\lambda p$ we get 
-\begin{equation}
-\begin{array}{lll}
-\tilde{r}_2([\lambda x_0\xi_0:\dots:\lambda x_n\xi_n], t)&=&\left[\sqrt[d]{(1-t)\lambda^d x^d_0+t\frac{\lambda^d x^d_0\chi_0}{1-\sum \lambda^d x^d_j(1-\chi_j)}}\xi_0:\dots:\sqrt[d]{(1-t)\lambda^d x^d_n+t\frac{\lambda^d x^d_n\chi_n}{1-\sum \lambda^d x^d_j(1-\chi_j)}}\xi_n\right]\\
-&=&\left[\sqrt[d]{(1-t)x^d_0+t\frac{x^d_0\chi_0}{1-\sum x^d_j(1-\chi_j)}}\xi_0:\dots:\sqrt[d]{(1-t)x^d_n+t\frac{x^d_n\chi_n}{1-\sum x^d_j(1-\chi_j)}}\xi_n\right]\\
-\end{array}
-\end{equation}
-
-=======
 \begin{equation*}
 \tilde{r}_2([x_0\xi_0:\dots:x_n\xi_n], t) =\left[H(x_0,\chi_0)\xi_0:\dots:H(x_n,\chi_n) \xi_n\right]
 \end{equation*}
@@ -476,7 +461,6 @@ Then, for the point $\lambda p$ we get
 \tilde{r}_2([\lambda x_0\xi_0:\dots:\lambda x_n\xi_n], t)&=\left[H(\lambda x_0, \chi_0)\xi_0:\dots:H(\lambda x_n, \chi_n)\xi_n\right]\\ 
 &=\left[H( x_0, \chi_0)\xi_0:\dots:H(x_n, \chi_n)\xi_n\right].
 \end{align*}
->>>>>>> 2a87fee69dfb1e6fdd6e608f9b4ee957cc3f5a74
 So we see that $\tilde{r}_2$ also takes both representatives will end to the same point.
 \end{proof}
 
@@ -514,17 +498,10 @@ First, we give a detailed description of $\Delta$. We will denote the $0$-simpli
 	X & =  [v_0,\dots,v_n]
 \end{align*}
 The face maps of $X$ and $X^{(j_0,\dots,j_i)}$ are:
-<<<<<<< HEAD
-\begin{equation}
-	\delta^i_k(X^{(j_0,\dots,j_i)}) =[v_{j_0},\dots,\hat{v},\dots,v_{j_i}] 
-\end{equation}
-where $\hat{v}$ indicates that we take out the vertix in position $k$, for $k\leq i$.
-=======
 \begin{equation*}
 	\delta^i_k(X^{(j_0,\dots,j_i)}) =[v_{j_0},\dots,\hat{v},\dots,v_{j_i}]
 \end{equation*}
 where $\hat{v}$ indicates that we take out the vertex in position $k$, for $k\leq i$.
->>>>>>> 2a87fee69dfb1e6fdd6e608f9b4ee957cc3f5a74
 an the boundary maps are given by:
 \begin{equation*}
 \partial^i =\sum (-1)^{k}\delta^i_k
@@ -553,20 +530,10 @@ X_{(1,\dots,\xi_i, 1, 1)}\cap\{x_j=1,\,\forall j\neq i\}.
 \]
 We can remark that $X_{(\xi_0, \dots, \xi_n)}=(\xi_0, \dots, \xi_n)\cdot X_{(1,\dots,1)}$, and therefore that we have a group action of $\mu^{n+1}_d$ over $S_d$.
 Let
-<<<<<<< HEAD
-\begin{equation}
-\begin{array}{rll}
-	X^{(j_0,\dots,j_i)}_{(1,\dots,1)} & = & X_{(1, \dots, 1)}\cap\{x_{j_k}\neq0 \forall k=0,\dots i\text{ and }x_{i}=0\text{ for the rest}\}\\
-	X^j_{(1,\dots,1)} & = & X_{(1, \dots, 1)}\cap\{x_i=0\forall i\neq j\}\\
-	
-\end{array}
-\end{equation}
-=======
 \begin{align*}
 	X^{(j_0,\dots,j_i)}_{(1,\dots,1)} & =  X_{(1, \dots, 1)}\cap\{x_{j_k}\neq0,\, \forall k=0,\dots i\text{ and }x_{i}=0\text{ for the rest}\}\\
 	X^j_{(1,\dots,1)} & =  X_{(1, \dots, 1)}\cap\{x_i=0,\,\forall i\neq j\}\\
 \end{align*}
->>>>>>> 2a87fee69dfb1e6fdd6e608f9b4ee957cc3f5a74
 We can describe the chain complexes. They are abelian groups but they are also $R_d$-complexes
 where $R_d$ is the group algebra $\ZZ[\mu_d^{n+1}]$. For the $n$-chains we have
 \[
@@ -795,31 +762,34 @@ 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 $\Delta^d$.
 \end{example}
 
-\begin{thebibliography}{99}
+\bibliographystyle{amsplain}
+\bibliography{biblio}
 
-\bibitem{andreotti1959}
+%\begin{thebibliography}{99}
-A. Andreotti and T. Frankel,
+%
-\textit{The Lefschetz theorem on hyperplane sections},
+%\bibitem{andreotti1959}
-Annals of Mathematics,
+%A. Andreotti and T. Frankel,
-Vol. 69, No. 3,
+%\textit{The Lefschetz theorem on hyperplane sections},
-pp. 713--717,
+%Annals of Mathematics,
-1959.
+%Vol. 69, No. 3,
-
+%pp. 713--717,
-\bibitem{libgober1986}
+%1959.
-A. Libgober,
+%
-\textit{On the homotopy type of the complement to plane algebraic curves},
+%\bibitem{libgober1986}
-J. Reine Angew. Math.,
+%A. Libgober,
-Vol. 367,
+%\textit{On the homotopy type of the complement to plane algebraic curves},
-pp. 103--114,
+%J. Reine Angew. Math.,
-1986.
+%Vol. 367,
-DOI: 10.1515/crll.1986.367.103
+%pp. 103--114,
-
+%1986.
-\end{thebibliography}
+%DOI: 10.1515/crll.1986.367.103
+%
+%\end{thebibliography}
 
 \appendix
 \section{SageMath code}
 
-We have written a SageMath function allowing us to define the $\Delta$-complex associated to the complement of a projective Fermat hypersurface of degree $d$ in $\PP^n$.
+We have written a SageMath~\cite{Sagemath} function allowing us to define the $\Delta$-complex associated to the complement of a projective Fermat hypersurface of degree $d$ in $\PP^n$.
 
 The code for the function is:
 \begin{python}