diff --git a/biblio.bib b/biblio.bib new file mode 100644 index 0000000..c8fb3cc --- /dev/null +++ b/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}, +} diff --git a/generaldegreeHypersurfaces.tex b/generaldegreeHypersurfaces.tex index bae4679..e77aab7 100644 --- a/generaldegreeHypersurfaces.tex +++ b/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 $n$. - -The retraction is performed in several steps, using the branched cover structure of the Fermat hypersurfaces to the degree $1$ case. +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. \end{abstract} \maketitle -\section{Definition of the problem} -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 $\CC\PP^2$). +\section*{Introduction} +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} -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} +%\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} \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}