From e0c4a4604e83efc32a55f321ca089c617b68d228 Mon Sep 17 00:00:00 2001 From: Alba Larraya Sancho Date: Thu, 24 Sep 2026 15:42:04 +0200 Subject: [PATCH] cambio en un dibujo --- generaldegreeHypersurfaces.tex | 91 ++++++++++++++++------------------ 1 file changed, 43 insertions(+), 48 deletions(-) diff --git a/generaldegreeHypersurfaces.tex b/generaldegreeHypersurfaces.tex index d8fe6c6..67c9453 100644 --- a/generaldegreeHypersurfaces.tex +++ b/generaldegreeHypersurfaces.tex @@ -32,7 +32,7 @@ decorations.markings, calc} %opening \title[Retraction of the complement of a smooth curve]{Retraction of the complement of smooth projective hypersurfaces to a $n$-dimensional $\Delta$-complex.} -\author[E. Artal]{Enrique Artal Bartolo} +\author[E. Artal]{Enrique Artal Bartolo} \address{Departamento de Matem\'{a}ticas, IUMA \\ Universidad de Zaragoza \\ C.~Pedro Cerbuna 12, 50009, Zaragoza, Spain} @@ -70,7 +70,7 @@ The retraction is performed in several steps, using the branched cover structure \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 $\CC\PP^2$). -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 problem for a general smooth hypersurface. +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. \section{Smooth hypersurfaces} @@ -97,7 +97,7 @@ We then have a commutative diagram: % K & \longhookrightarrow & \PP^n\setminus C_d \\ % \end{array} \begin{tikzcd} -S_d\rar[hookrightarrow]\dar["\proy{}"]& M_d\dar["\proy{}"]\\ +S_d\rar[hookrightarrow]\dar["\proy{}"]& M_d\dar["\proy{}"]\\ K\rar[hookrightarrow] & \PP^n\setminus C_d. \end{tikzcd} \end{equation*} @@ -119,7 +119,7 @@ i.e. $x^r,x^i\in\RR$ and $x=x^r + i x^i$. This notation also applies to vectors % M_1 & \longrightarrow & M_1\cap\RR^3&\longrightarrow&S_1 \\ % \end{array} %\end{equation*} -We will now define a strong retraction map +We will now define a strong retraction map \begin{equation} \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\\ @@ -140,7 +140,7 @@ Note that for $(x_0,\dots,x_n)\in M_1$, we already have that \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 $(x_0,\dots,x_n)\in M_1$. +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 $(x_0,\dots,x_n)\in M_1$. Let us summarize it. @@ -155,21 +155,21 @@ The map $r_1$ is a strong deformation retraction from $M_1$ to $M_1\cap\RR^{n+1} Now we will define a strong deformation retraction \[ -r_2:(M_1 \cap \RR^{n+1})\times[0, 1]\to M_1 \cap \RR^{n+1} +r_2:(M_1 \cap \RR^{n+1})\times[0, 1]\to M_1 \cap \RR^{n+1} \] from $M_1 \cap \RR^{n+1}$ to $S_1$. Note that $M_1\cap\RR^{n+1}$ -is a hypersurface in $\RR^{n+1}$, in Figure~\ref{fig:triangulo} we can see the example $n=2$. +is a hypersurface in $\RR^{n+1}$, in Figure~\ref{fig:triangulo} we can see the example $n=2$. \begin{lemma} The space $S_1$ is a $n$-simplex. \end{lemma} \begin{proof} -Let $e_i=(0,\dots,1,\dots,0)$ be the vectors of the canonical base. Then +Let $e_i=(0,\dots,1,\dots,0)$ be the vectors of the canonical base. Then \begin{equation*} \begin{array}{lllll} S_1&=&\{(x_0,\dots,x_n)\in\RR^{n+1}|\sum x_i=1, x_i\geq1\}&& \\ -&=&\{x_0e_0+\dots+x_ne_n|\sum x_i=1, x_i\geq1\}&=&\Delta_n\\ +&=&\{x_0e_0+\dots+x_n|\sum x_i=1, x_i\geq1\}&=&\Delta_n\\ \end{array} \qedhere \end{equation*} @@ -227,13 +227,13 @@ S_1&=&\{(x_0,\dots,x_n)\in\RR^{n+1}|\sum x_i=1, x_i\geq1\}&& \\ \end{center} \end{figure} -As we can see in Figure \ref{fig:triangulo}, the hyperplanes defined by $\{x_i=0\}$ divide the space $S_1$ in $2^{n+1}-1$ regions. We denote each of these regions as $(\pm,\dots,\pm)$ indicating which coordinates are greater or equal to $0$ and which are lower or equal to $0$. For example, the triangle corresponding to $S_1$ would be $(+,\dots,+)$. -Note that there is no region $(-,\dots,-)$. +As we can see in Figure \ref{fig:triangulo}, the hypersurfaces defined by $\{x_i=0\}$ divide the space in $2^{n+1}-1$ sections. We denote each of these sections as $(\pm,\dots,\pm)$ indicating which coordinates are greater or equal to $0$ and which are lower or equal to $0$. For example, the triangle corresponding to $S_1$ would be $(+,\dots,+)$. +Note that there is no section $(-,\dots,-)$. The retraction in $(+,\dots,+)$ is the identity for any value of $t$. For the rest of the parts, we define a retraction: \begin{equation} \begin{tikzcd}[row sep=0pt,/tikz/column 1/.append style={anchor=base east}, /tikz/column 2/.append style={anchor=base west}] -(M_1\cap\RR^{n+1})\times{[0,1]}\rar["r_2"]&M_1\cap\RR^{n+1}\\ +(M_1\cap\RR^{n+1})\times{[0,1]}\rar["r_2"]&M_1\cap\RR^{n+1}\times [0,1]\\ (Q, t)\rar[mapsto]& t P_Q + (1 - t) Q, \end{tikzcd} \end{equation} @@ -243,7 +243,7 @@ The retraction in $(+,\dots,+)$ is the identity for any value of $t$. For the re % (x,y,z, t) & \longrightarrow & (1-t)(x,y,z)+tP_{x,y,z} \\ % \end{array} %\end{equation} -where $P_Q$ is a point depending on $Q\in M_1\cap\RR^{n+1}$. +where $P_Q$ is a point depending on $Q\in M_1\cap\RR^{n+1}$. In general, for a point $Q$ in a region $(\pm,\dots,\pm)$, we want the retraction to take the negative coordinates to $0$ and to change the positive ones such that at the end the sum is $1$. Therefore, we can write the point $P_Q$ as: \begin{equation} @@ -315,20 +315,15 @@ We can divide $\CC$ in $2d$ sectors of angle $\pi/d$ as in figure \ref{divisionC \begin{figure}[h] \begin{center} \begin{tikzpicture}[scale=0.6] + \draw (-5,0) to (5,0); - \draw (5,1.5) to (-5,-1.5); - \draw (5,3.2) to (-5, -3.2); - \draw (4.7,5) to (-4.7,-5); \draw (2.8,5) to (-2.8,-5); - \draw (0.9,5) to (-0.9,-5); - \draw (-0.9,5) to (0.9,-5); - \draw (-2.8,5) to (2.8,-5); \draw (-4.7,5) to (4.7,-5); - \draw (-5,3.2) to (5, -3.2); - \draw (-5,1.5) to (5,-1.5); + \draw[->] (3,0) to[out=90, in=350] (1,2) node[ xshift=1.0cm, yshift=-0.2cm]{$\pi/d$}; + \end{tikzpicture} \end{center} -\caption{Division of the complex plane} +\caption{Division of the complex plane when $d=3$} \label{divisionC} \end{figure} @@ -380,8 +375,8 @@ Then, we can see that $G(z,t)^d=\bar{r}(z^d,t)$, since: We can the write the lifting of $r_1$ as $\tilde{r}_1((x_0,\dots,x_n),t)=(G(x,t), \dots, G(x_n,t))$. -Lastly, we are going to see how to lift $r_2$ in order to get a retraction from $M_d\cap\{x^d_0, \ldots , x^d_n\in\RR\}$ into $S_d$. -For a point $P=(x_0\xi_0,\dots,x_n\xi_n)\in M_d\cap\{x^d_0, \ldots , x^d_n\in\RR\}$ (where $(x_0,\dots,x_n)$ are real and $\xi_i$ are $d$-th roots of unity), the corresponding point in $M_1\cap\RR^{n+1}$ is $(x^d_0,\dots,x^d_n)$ and therefore the goal point of the retraction $r_2$ will be +Lastly, we are going to see how to lift $r_2$ in order to get a retraction from $M_d\cap\{x^d_0, y^d, x^d_n\in\RR\}$ into $S_d$. +For a point $P=(x_0\xi_0,\dots,x_n\xi_n)\in M_d\cap\{x^d_0, y^d, x^d_n\in\RR\}$ (where $(x_0,\dots,x_n)$ are real and $\xi_i$ are $d$-th roots of unity), the corresponding point in $M_1\cap\RR^{n+1}$ is $(x^d_0,\dots,x^d_n)$ and therefore the goal point of the retraction $r_2$ will be \begin{equation} P^d_Q=\left(\frac{x^d_0\chi_0}{1-\sum x^d_j(1-\chi_j)}, \dots,\frac{x^d_n\chi_n}{1-\sum x^d_j(1-\chi_j)} \right) \end{equation} @@ -427,7 +422,7 @@ We have then proved that $S_d$ is a strong deformation retract of $M_d$. Since $\lambda$ is a $d-$th root of unity, we get that $G(x,t)=G(\lambda x,t)$. Therefore, we get that $\tilde{r}_1(p,t)=\tilde{r}_1(\lambda p,t)$. It's just left to check that $r_2$ also sends them to the same class. 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 \begin{equation} \begin{array}{lll} @@ -436,7 +431,7 @@ We have then proved that $S_d$ is a strong deformation retract of $M_d$. \end{equation} -Then, for the point $\lambda p$ we get +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]\\ @@ -461,8 +456,8 @@ We denote as $\Delta$ the $\Delta-$complex $[v_0, \dots, v_n]$ that has $n+1$ ve % & & K & \hookrightarrow & \PP^2\setminus C_d \\ % \end{array} \begin{tikzcd} -\Delta \rar[leftrightarrow]& S_1&\\ -\Delta^d\uar\rar[leftrightarrow]&S_d\uar\rar[hookrightarrow]\dar& M_d\dar\\ +\Delta \rar[leftrightarrow]& S_1&\\ +\Delta^d\uar\rar[leftrightarrow]&S_d\uar\rar[hookrightarrow]\dar& M_d\dar\\ &K \rar[hookrightarrow]& \PP^n\setminus C_d, \end{tikzcd} \end{equation*} @@ -485,7 +480,7 @@ First, we give a detailed description of $\Delta$. We will denote the $0$-simpli The face maps of $X$ and $X^{(j_0,\dots,j_i)}$ are: \begin{equation} - \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} where $\hat{v}$ indicates that we take out the vertix in position $k$, for $k\leq i$. an the boundary maps are given by: @@ -506,11 +501,11 @@ Let $\mu_d=\{\xi| \xi^d=1\}$. We can express $S_d$ as a union \end{equation} where $X_{(\xi_0, \dots, \xi_n)}=\{(x_0,\dots,x_n)| \sum x^d_i=1 \text{ and } \xi_0x, \dots, \xi_nx_n\geq0\}$. -Then, we can see that the faces of the simplex are given by $\mathring{X}_{(\xi_0, \dots, \xi_n)}$, while the $j-$simplices are given by +Then, we can see that the faces of the simplex are given by $\mathring{X}_{(\xi_0, \dots, \xi_n)}$, while the $j-$simplices are given by \[ X_{(\xi_0, \dots,\xi_n)}\cap\{x_{i_1},\dots,x_{i_j}=0\}, \] - where $\xi_{i_1},\dots,\xi_{i_j}=1$, and the vertices by + where $\xi_{i_1},\dots,\xi_{i_j}=1$, and the vertices by \[ X_{(1,\dots,\xi_i, 1, 1)}\cap\{x_j=1\forall j\neq i\}. \] @@ -520,7 +515,7 @@ Let \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} We can describe the chain complexes. They are abelian groups but they are also $R_d$-complexes @@ -530,7 +525,7 @@ C_{n}(\Delta^d) = R_d\langle X_{(1,\dots,1)}\rangle, \] which is a free principal $R_d$-module, i.e. a free abelian group of rank~$d^{n+1}$. For the $i$-chains, with $i\\ %C_1(\Delta^d) & = & \ZZ[\mu^3_d]/\ZZ[\mu_d] \bigoplus \ZZ[\mu^3_d]/\ZZ[\mu_d] \bigoplus \ZZ[\mu^3_d]/\ZZ[\mu_d]\\ @@ -548,7 +543,7 @@ a free abelian group of rank~$(n+1)d$. %\end{array} %\end{equation} -The face maps are $R_d$-morphisms defined as +The face maps are $R_d$-morphisms defined as \begin{equation} \begin{aligned} \delta^n_i(X_{(1,1,1)}) & =X^{(0,\dots,i-1,i+1,\dots,n)}_{(1,\dots,1)} &&& \delta^j_i(X^{(k_0,\dots,k_j)}_{(1,\dots,1)})&= X^{(k_0,\dots,k_{i-1},k_{i+1},\dots,k_n)}_{(1,\dots,1)} . @@ -588,8 +583,8 @@ We have then defined $\Delta^d$, the simplicial complex with which we can identi % & & K & \hookrightarrow & \PP^2\setminus C_d \\ % \end{array} \begin{tikzcd} -\Delta \rar[leftrightarrow]& S_1&\\ -\Delta^d\uar\rar[leftrightarrow]\dar&S_d\uar\rar[hookrightarrow]\dar& M_d\dar\\ +\Delta \rar[leftrightarrow]& S_1&\\ +\Delta^d\uar\rar[leftrightarrow]\dar&S_d\uar\rar[hookrightarrow]\dar& M_d\dar\\ \Delta^d_\textrm{proj}\rar[leftrightarrow]&K \rar[hookrightarrow]& \PP^2\setminus C_d, \end{tikzcd} \end{equation*} @@ -604,10 +599,10 @@ We have then defined $\Delta^d$, the simplicial complex with which we can identi %\end{equation*} The map $\Delta^d\longrightarrow\Delta^d_\text{proj}$ is given by $(x_0,\dots,x_n)\longrightarrow[x_0:\dots:x_n]$. Since we are passing to the projective plane, two points $(x_0\xi_0, \dots, x_n\xi_n)$ and $(x_0\xi_0\xi, \dots, x_n\xi_n\xi)$ will have the same image. -The $n$-cells $X_{(\xi_0, \dots, \xi_n)}$ and $X_{(\xi_0\xi, \dots, \xi_n\xi)}$ of $\Delta^d$, project the same $n$-cell of $\Delta^d_\textrm{proj}$, which will be denoted as $X_{[\xi_0: \dots: \xi_n]}$. +The $n$-cells $X_{(\xi_0, \dots, \xi_n)}$ and $X_{(\xi_0\xi, \dots, \xi_n\xi)}$ of $\Delta^d$, project the same $n$-cell of $\Delta^d_\textrm{proj}$, which will be denoted as $X_{[\xi_0: \dots: \xi_n]}$. The same convention is followed for the rest of $i-$simplices. Note that for the vertices, since all coordinates but one are $0$, we have only $n+1$ vertices $X^0, \dots, X^n$. -The group $\mu_d^{n+1}$ also acts, but the action of the diagonal vectors is trivial. Let us denote +The group $\mu_d^{n+1}$ also acts, but the action of the diagonal vectors is trivial. Let us denote \[ T_d:=\ZZ\left[\frac{\mu_d^{n+1}}{\mu_d\langle(\xi,\dots,\xi)\rangle}\right]; \] @@ -615,7 +610,7 @@ we use variables $t_0,\dots,t_n$ as for $R_d$ and we have to add the relation $t We have the following chain complexes: \begin{align*} C_n(\Delta^d_\text{proj}) &= T_d\langle X_{[1:\dots:1]}\rangle,\\ -C_i(\Delta^d_\text{proj}) &= \bigoplus_{(t_{j_1},\dots,t_{j_{n-i}})}\frac{T_d}{(t_{j_1} - 1,\dots,t_{j_{n-i}} - 1)} \langle X^{(k_0,\dots,k_i)}_{[1:\dots:1]}\rangle,\\ +C_i(\Delta^d_\text{proj}) &= \bigoplus_{(t_{j_1},\dots,t_{j_{n-i}})}\frac{T_d}{(t_{j_1} - 1,\dots,t_{j_{n-i}} - 1)} \langle X^{(k_0,\dots,k_i)}_{[1:\dots:1]}\rangle,\\ C_0(\Delta^d) &= \bigoplus_{t_j}\frac{T_d}{(t_0 - 1,\dots,t_{j-1}-1,t_{j+1}-1,\dots, t_n - 1)} \langle X^j_{[1:\dots:1]}\rangle. \end{align*} %As for the points where one coordinate is $0$, say, for example the first one, since the group action is then by $\mu^3_2/\mu_2$, we write the elements of $\mu^3_2/\mu_2$ as $(1,\xi_1,\xi_2)$, where the $1$ is on the coordinate that is $0$. Then, passing to the projective space, we will say that $L^x_{[1:\xi_1,\xi_2]}$ and $L^x_{[1:\xi'_1,\xi'_2]}$ are the same if $[1:\xi'_1,\xi'_2]=[1:\xi_1\xi,\xi_2\xi]$ for some $\xi\in\mu_2$. The rest of $1-$simplices would be identified similarly. Lastly, for the $0-$simplices, since we have $2$ zero coordinates, we have that all $0-$simplices with the same nonzero coordinate will be identified to each other. As a result, $\Delta^d_\text{proj}$ will only have $3$ $0-$simplices, $3d$ $1-$simplices and $d^2$ $2-$simplices. @@ -782,7 +777,7 @@ The code for the function is: def DeltaComplex_hypersurface(n,d): resdic = [{tuple(i*[0]+[1] + (n-i)*[0]): None for i in range(n+1) }]+ [{} for i in range(n)] - for nzerosi in range(n+1): + for nzerosi in range(n+1): for t in Tuples(range(d+1),n-nzerosi ): T = nzerosi*[0] + [1] + list(t) dimT = len(T) - T.count(0) - 2 @@ -795,9 +790,9 @@ def DeltaComplex_hypersurface(n,d): canonicalize(CT) caras.append(tuple(CT)) resdic[dimT+1][tuple(T)] = caras - + DC = [[tuple() for _ in range(n+1)]] - for i in range(1,n+1): + for i in range(1,n+1): lis = [] for p,v in resdic[i].items(): lisp = [] @@ -805,15 +800,15 @@ def DeltaComplex_hypersurface(n,d): lisp.append(list(resdic[i-1].keys()).index(s)) lis.append(lisp) DC.append(lis) - - dc=DeltaComplex(DC) + + dc=DeltaComplex(DC) return dc \end{python} -This function uses an auxiliary function, named $canonicalize$ in the code. +This function uses an auxiliary function, named $canonicalize$ in the code. $canonicalize$ is used to identify the simplices in the projective space. The code to this function is: \begin{python} -def canonicalize(t): +def canonicalize(t): v = next(filter(lambda a : a != 0, t)) correct = v - 1 for i in range(len(t)): @@ -848,7 +843,7 @@ Delta_Complex = DeltaComplex_hypersurface(4,6) Once we have gotten the $\Delta$-complex, we can compute the fundamental group or the cohomology of the space, for example. \begin{python} Dc = SimplicialSet(Delta_Complex) -dc_punto = Dc.set_base_point(Dc.all_n_simplices(0)[0]) +dc_punto = it commit Dc.set_base_point(Dc.all_n_simplices(0)[0]) dc_punto.fundamental_group() -> Finitely presented group < e0 | e0^6 > dc_punto.cohomology()