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generaldegreeHypersurfaces.tex
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@ -81,16 +81,16 @@ This retraction is performed in several steps, using the branched cover structur
\section*{Introduction} \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 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. 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. However, no general method has been given to explicitly compute 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. 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.
Possibly the only known method applies to the following particular case. Possibly the only known method applies to the following particular case.
%One main problem is to provide such a CW-complex. %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). 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 in $\CC^2$ 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 $>1$. 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. 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 case 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 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 $[x_0:\dots:x_n]\mapsto[x_0^d:\dots:x_n^d]$. The main point is to understand the lifting of this complex by the Kummer cover $[x_0:\dots:x_n]\mapsto[x_0^d:\dots:x_n^d]$.
@ -146,7 +146,7 @@ We will now define a strong retraction map
\begin{equation} \begin{equation}
\begin{tikzcd}[row sep=0pt,/tikz/column 1/.append style={anchor=base east}, /tikz/column 2/.append style={anchor=base west}] \begin{tikzcd}[row sep=0pt,/tikz/column 1/.append style={anchor=base east}, /tikz/column 2/.append style={anchor=base west}]
M_1\times{[0,1]}\rar["r_1"]&M_1\\ M_1\times{[0,1]}\rar["r_1"]&M_1\\
(P, t)\rar[mapsto]& t P_r + (1 - t) P_i=:r_{1t}(P); (P, t)\rar[mapsto]& t P^r + (1 - t) P^i=:r_{1t}(P);
\end{tikzcd} \end{tikzcd}
\end{equation} \end{equation}
note that $r_{10}$ is the identity,$r_{11}(M_1)=M_1\cap\RR^{n+1}$ note that $r_{10}$ is the identity,$r_{11}(M_1)=M_1\cap\RR^{n+1}$
@ -502,7 +502,7 @@ The face maps of $X$ and $X^{(j_0,\dots,j_i)}$ are:
\delta^i_k(X^{(j_0,\dots,j_i)}) =[v_{j_0},\dots,\hat{v},\dots,v_{j_i}] \delta^i_k(X^{(j_0,\dots,j_i)}) =[v_{j_0},\dots,\hat{v},\dots,v_{j_i}]
\end{equation*} \end{equation*}
where $\hat{v}$ indicates that we take out the vertex in position $k$, for $k\leq i$. where $\hat{v}$ indicates that we take out the vertex in position $k$, for $k\leq i$.
an the boundary maps are given by: And the boundary maps are given by:
\begin{equation*} \begin{equation*}
\partial^i =\sum (-1)^{k}\delta^i_k \partial^i =\sum (-1)^{k}\delta^i_k
\label{bordermaps} \label{bordermaps}
@ -642,6 +642,7 @@ C_0(\Delta^d) &= \bigoplus_{t_j}\frac{T_d}{(t_0 - 1,\dots,t_{j-1}-1,t_{j+1}-1,\d
%\end{equation} %\end{equation}
\begin{example}{} \begin{example}{}
We are going to compute now the simplicial complex for $S_d$ and $K$ when $d=n=2$, this is, our curve will be $C_D=\{[x:y:z]\in\PP^2|x^2+y^2+z^2=0\}$, $S_2=\{(x,y,z)\in M_2| x^2, y^2, z^2\in\RR_{\geq 0}\}$ and $K=\{[x:y:z]\in\PP^2|x^2,y^2,z^2\in\RR_{\geq0}\}$. We are going to compute now the simplicial complex for $S_d$ and $K$ when $d=n=2$, this is, our curve will be $C_D=\{[x:y:z]\in\PP^2|x^2+y^2+z^2=0\}$, $S_2=\{(x,y,z)\in M_2| x^2, y^2, z^2\in\RR_{\geq 0}\}$ and $K=\{[x:y:z]\in\PP^2|x^2,y^2,z^2\in\RR_{\geq0}\}$.
We start with $S_2$. First, we can observe that, from all points in $\CC$, the only ones whose square power is real are either the real numbers of the pure imaginary numbers. And from these, the later have negative square power. Therefore, we can conclude that $S_2=\{(x,y,z)\in\RR^3| x^2+y^2+z^2=1\}$, that is, $S_2$ is the real three dimensional sphere. We now see the simplicial complex $\Delta^2$ that can be associated to it such that $\phi(\Delta^2)=\Delta$. We start with $S_2$. First, we can observe that, from all points in $\CC$, the only ones whose square power is real are either the real numbers of the pure imaginary numbers. And from these, the later have negative square power. Therefore, we can conclude that $S_2=\{(x,y,z)\in\RR^3| x^2+y^2+z^2=1\}$, that is, $S_2$ is the real three dimensional sphere. We now see the simplicial complex $\Delta^2$ that can be associated to it such that $\phi(\Delta^2)=\Delta$.
@ -671,7 +672,66 @@ The face maps of $L^z_{(1,1,1)}$, $L^y_{(1,1,1)}$ and $L^x_{(1,1,1)}$ are:
\delta^1_1(L^x_{(1,1,1)}) \rar[equal] & V^y_{(1,1,1)} \rar[equal] & (0,0,1)\\ \delta^1_1(L^x_{(1,1,1)}) \rar[equal] & V^y_{(1,1,1)} \rar[equal] & (0,0,1)\\
\end{tikzcd} \end{tikzcd}
\end{equation*} \end{equation*}
Lastly, the $0-$simplices will be $V^x_{(1,1,1)}=(1,0,0)$, $V^x_{(-1,1,1)}=(-1,0,0)$, $V^y_{(1,1,1)}=(0,1,0)$, $V^y_{(1,-1,1)}=(0,-1,0)$, $V^z_{(1,1,1)}=(0,0,1)$ and $V^z_{(1,1,-1)}=(0,0,-1)$. In Figure \ref{fig:S2} we can see the $\Delta-$complex that we associate to $S_2$ which corresponds to the partition of the $2$-spheres in triangles with right angles. Lastly, the $0-$simplices will be $V^x_{(1,1,1)}=(1,0,0)$, $V^x_{(-1,1,1)}=(-1,0,0)$, $V^y_{(1,1,1)}=(0,1,0)$, $V^y_{(1,-1,1)}=(0,-1,0)$, $V^z_{(1,1,1)}=(0,0,1)$ and $V^z_{(1,1,-1)}=(0,0,-1)$. In Figure \ref{fig:S2} we can see the $\Delta-$complex that we associate to $S_2$ which corresponds to the partition of the $2$-spheres in triangles with right angles.
\begin{figure}[!hp]
\begin{center}
\begin{tikzpicture}[scale=0.7]
\draw[gray, thin] (-5,0) to (5,0);
\draw[gray, thin] (0,-5) to (0,5);
\draw[gray, thin] (-4,-4) to (4,4);
\draw[fill, red] (4,0) circle (2pt);
\draw[fill, red] (-4,0) circle (2pt);
\draw[fill, green] (0,4) circle (2pt);
\draw[fill, green] (0,-4) circle (2pt);
\draw[fill, blue] (1.41,1.41) circle (2pt);
\draw[fill, blue] (-1.41,-1.41) circle (2pt);
\draw (0,0) circle (4cm);
\draw (-4,0) arc (180:360:4 and 1.5);
\draw[dashed] (4,0) arc (0:180:4 and 1.5);
\draw[dashed] (0,4) arc (90:270:-1.5 and 4);
\draw (0,-4) arc (270:90:1.5 and 4);
\draw[pink, thick](-4,0.02) arc (180:250:4 and 1.5);
\draw[pink, thick](-4.02,0.02) arc (180:90:4);
\draw[pink, thick](0,4) arc (270:90:1.5 and -4);
\draw[purple, thick](-4,-0.02) arc (180:250:4 and 1.5);
\draw[purple, thick](-4,-0.02) arc (180: 270:4);
\draw[purple, thick] (-1.43,-1.41) arc (200:270:1.5 and 3.9);
\draw[yellow, thick, dashed](-3.98,0.02) arc (180:70:4 and 1.5);
\draw[yellow, thick, dashed](-3.98,0.02) arc (180: 90:4);
\draw[yellow, thick, dashed] (1.39,1.41) arc (200:270:-1.5 and -3.9);
\draw[orange, thick, dashed](-3.98,-0.02) arc (180:70:4 and 1.5);
\draw[orange, thick, dashed](-3.98,-0.02) arc (180: 270:4);
\draw[orange, thick, dashed] (1.39,1.41) arc (160:270:-1.5 and 4);
\draw[teal, thick, dashed](4.02,0.02) arc (180:110:-4 and 1.5);
\draw[teal, thick, dashed](4.02,0.02) arc (180: 270:-4);
\draw[teal, thick, dashed] (1.42,1.41) arc (200:270:-1.5 and -3.9);
\draw[violet, thick](4.02,-0.02) arc (0:110:4 and -1.5);
\draw[violet, thick](4.02,-0.02) arc (180: 90:-4);
\draw[violet, thick] (-1.39,-1.42) arc (200:270:1.5 and 3.9);
\draw[magenta, thick, dashed](3.98,0.02) arc (180:110:-4 and 1.5);
\draw[magenta, thick, dashed](3.98,-0.02) arc (180: 90:-4);
\draw[magenta, thick, dashed] (1.43,1.41) arc (160:270:-1.5 and 4);
\draw[lime, thick, dashed](4.02,0.02) arc (0:110:4 and -1.5);
\draw[lime, thick, dashed](3.98,0.02) arc (180: 270:-4);
\draw[lime, thick, dashed] (0.02,3.98) arc (270:160:1.5 and -4);
\end{tikzpicture}
\end{center}
\caption{Division in simplices of $S_2$}
\label{fig:S2}
\end{figure}
%The vertices of $\Delta$, $v_0$, $v_1$ and $v_2$, can be identified to $(1,0,0)$, $(0,0,1)$ and $(0,1,0)$ respectively. The preimages of these points are: %The vertices of $\Delta$, $v_0$, $v_1$ and $v_2$, can be identified to $(1,0,0)$, $(0,0,1)$ and $(0,1,0)$ respectively. The preimages of these points are:
%\begin{equation} %\begin{equation}
%\begin{array}{rll} %\begin{array}{rll}
@ -694,63 +754,6 @@ Lastly, the $0-$simplices will be $V^x_{(1,1,1)}=(1,0,0)$, $V^x_{(-1,1,1)}=(-1,0
%Finally, we will have $8$ $2$-faces. For three edges to form a face on the simplex, we need the corresponding vertices of the edges to coincide, that is, the edges with non-zero first coordinate must both take the same sign on said coordinate, and equivalently for the rest of coordinates. Then, the faces are defined by taking the square root of the coordinates of the points in $S_1$ with sign equal to those in the edges. %Finally, we will have $8$ $2$-faces. For three edges to form a face on the simplex, we need the corresponding vertices of the edges to coincide, that is, the edges with non-zero first coordinate must both take the same sign on said coordinate, and equivalently for the rest of coordinates. Then, the faces are defined by taking the square root of the coordinates of the points in $S_1$ with sign equal to those in the edges.
\begin{figure}[!hp]
\begin{center}
\begin{tikzpicture}[scale=0.7]
\draw[gray, thin] (-5,0) to (5,0);
\draw[gray, thin] (0,-5) to (0,5);
\draw[gray, thin] (-4,-4) to (4,4);
\draw[fill, red] (4,0) circle (2pt);
\draw[fill, red] (-4,0) circle (2pt);
\draw[fill, green] (0,4) circle (2pt);
\draw[fill, green] (0,-4) circle (2pt);
\draw[fill, blue] (1.41,1.41) circle (2pt);
\draw[fill, blue] (-1.41,-1.41) circle (2pt);
\draw (0,0) circle (4cm);
\draw (-4,0) arc (180:360:4 and 1.5);
\draw[dashed] (4,0) arc (0:180:4 and 1.5);
\draw[dashed] (0,4) arc (90:270:-1.5 and 4);
\draw (0,-4) arc (270:90:1.5 and 4);
\draw[pink, thick](-4,0.02) arc (180:250:4 and 1.5);
\draw[pink, thick](-4.02,0.02) arc (180:90:4);
\draw[pink, thick](0,4) arc (270:90:1.5 and -4);
\draw[purple, thick](-4,-0.02) arc (180:250:4 and 1.5);
\draw[purple, thick](-4,-0.02) arc (180: 270:4);
\draw[purple, thick] (-1.43,-1.41) arc (200:270:1.5 and 3.9);
\draw[yellow, thick, dashed](-3.98,0.02) arc (180:70:4 and 1.5);
\draw[yellow, thick, dashed](-3.98,0.02) arc (180: 90:4);
\draw[yellow, thick, dashed] (1.39,1.41) arc (200:270:-1.5 and -3.9);
\draw[orange, thick, dashed](-3.98,-0.02) arc (180:70:4 and 1.5);
\draw[orange, thick, dashed](-3.98,-0.02) arc (180: 270:4);
\draw[orange, thick, dashed] (1.39,1.41) arc (160:270:-1.5 and 4);
\draw[teal, thick, dashed](4.02,0.02) arc (180:110:-4 and 1.5);
\draw[teal, thick, dashed](4.02,0.02) arc (180: 270:-4);
\draw[teal, thick, dashed] (1.42,1.41) arc (200:270:-1.5 and -3.9);
\draw[violet, thick](4.02,-0.02) arc (0:110:4 and -1.5);
\draw[violet, thick](4.02,-0.02) arc (180: 90:-4);
\draw[violet, thick] (-1.39,-1.42) arc (200:270:1.5 and 3.9);
\draw[magenta, thick, dashed](3.98,0.02) arc (180:110:-4 and 1.5);
\draw[magenta, thick, dashed](3.98,-0.02) arc (180: 90:-4);
\draw[magenta, thick, dashed] (1.43,1.41) arc (160:270:-1.5 and 4);
\draw[lime, thick, dashed](4.02,0.02) arc (0:110:4 and -1.5);
\draw[lime, thick, dashed](3.98,0.02) arc (180: 270:-4);
\draw[lime, thick, dashed] (0.02,3.98) arc (270:160:1.5 and -4);
\end{tikzpicture}
\end{center}
\caption{Division in simplices of $S_2$}
\label{fig:S2}
\end{figure}
Now, we give a description of $\Delta^2_\text{proj}$. First, for the $0-$simplices, we have that $V^x_{[1,1,1]}=V^x_{[-1,1,1]}=[1:0:0]$, $V^y_{[1,1,1]}=V^y_{[1,-1,1]}=[0:1:0]$ and $V^z_{[1,1,1]}=V^z_{[1,1,-1]}=[0:0:1]$. Now, we give a description of $\Delta^2_\text{proj}$. First, for the $0-$simplices, we have that $V^x_{[1,1,1]}=V^x_{[-1,1,1]}=[1:0:0]$, $V^y_{[1,1,1]}=V^y_{[1,-1,1]}=[0:1:0]$ and $V^z_{[1,1,1]}=V^z_{[1,1,-1]}=[0:0:1]$.