From 2a87fee69dfb1e6fdd6e608f9b4ee957cc3f5a74 Mon Sep 17 00:00:00 2001 From: Enrique Artal Date: Thu, 24 Sep 2026 14:49:16 +0200 Subject: [PATCH] falta abstract e introduccion --- .gitignore | 303 +++++++++++++++++++++++++++++++++ generaldegreeHypersurfaces.tex | 89 +++++----- 2 files changed, 344 insertions(+), 48 deletions(-) create mode 100644 .gitignore diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..4579efc --- /dev/null +++ b/.gitignore @@ -0,0 +1,303 @@ +# ---> TeX +## Core latex/pdflatex auxiliary files: +*.aux +*.lof +*.log +*.lot +*.fls +*.out +*.toc +*.fmt +*.fot +*.cb +*.cb2 +.*.lb + +## Intermediate documents: +*.dvi +*.xdv +*-converted-to.* +# these rules might exclude image files for figures etc. +# *.ps +# *.eps +*.pdf + +## Generated if empty string is given at "Please type another file name for output:" +.pdf + +## Bibliography auxiliary files (bibtex/biblatex/biber): +*.bbl +*.bcf +*.blg +*-blx.aux +*-blx.bib +*.run.xml + +## Build tool auxiliary files: +*.fdb_latexmk +*.synctex +*.synctex(busy) +*.synctex.gz +*.synctex.gz(busy) +*.pdfsync + +## Build tool directories for auxiliary files +# latexrun +latex.out/ + +## Auxiliary and intermediate files from other packages: +# algorithms +*.alg +*.loa + +# achemso +acs-*.bib + +# amsthm +*.thm + +# beamer +*.nav +*.pre +*.snm +*.vrb + +# changes +*.soc + +# comment +*.cut + +# cprotect +*.cpt + +# elsarticle (documentclass of Elsevier journals) +*.spl + +# endnotes +*.ent + +# fixme +*.lox + +# feynmf/feynmp +*.mf +*.mp +*.t[1-9] +*.t[1-9][0-9] +*.tfm + +#(r)(e)ledmac/(r)(e)ledpar +*.end +*.?end +*.[1-9] +*.[1-9][0-9] +*.[1-9][0-9][0-9] +*.[1-9]R +*.[1-9][0-9]R +*.[1-9][0-9][0-9]R +*.eledsec[1-9] +*.eledsec[1-9]R +*.eledsec[1-9][0-9] +*.eledsec[1-9][0-9]R +*.eledsec[1-9][0-9][0-9] +*.eledsec[1-9][0-9][0-9]R + +# glossaries +*.acn +*.acr +*.glg +*.glo +*.gls +*.glsdefs +*.lzo +*.lzs +*.slg +*.slo +*.sls + +# uncomment this for glossaries-extra (will ignore makeindex's style files!) +# *.ist + +# gnuplot +*.gnuplot +*.table + +# gnuplottex +*-gnuplottex-* + +# gregoriotex +*.gaux +*.glog +*.gtex + +# htlatex +*.4ct +*.4tc +*.idv +*.lg +*.trc +*.xref + +# hyperref +*.brf + +# knitr +*-concordance.tex +# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files +# *.tikz +*-tikzDictionary + +# listings +*.lol + +# luatexja-ruby +*.ltjruby + +# makeidx +*.idx +*.ilg +*.ind + +# minitoc +*.maf +*.mlf +*.mlt +*.mtc[0-9]* +*.slf[0-9]* +*.slt[0-9]* +*.stc[0-9]* + +# minted +_minted* +*.pyg + +# morewrites +*.mw + +# newpax +*.newpax + +# nomencl +*.nlg +*.nlo +*.nls + +# pax +*.pax + +# pdfpcnotes +*.pdfpc + +# sagetex +*.sagetex.sage +*.sagetex.py +*.sagetex.scmd + +# scrwfile +*.wrt + +# svg +svg-inkscape/ + +# sympy +*.sout +*.sympy +sympy-plots-for-*.tex/ + +# pdfcomment +*.upa +*.upb + +# pythontex +*.pytxcode +pythontex-files-*/ + +# tcolorbox +*.listing + +# thmtools +*.loe + +# TikZ & PGF +*.dpth +*.md5 +*.auxlock + +# titletoc +*.ptc + +# todonotes +*.tdo + +# vhistory +*.hst +*.ver + +# easy-todo +*.lod + +# xcolor +*.xcp + +# xmpincl +*.xmpi + +# xindy +*.xdy + +# xypic precompiled matrices and outlines +*.xyc +*.xyd + +# endfloat +*.ttt +*.fff + +# Latexian +TSWLatexianTemp* + +## Editors: +# WinEdt +*.bak +*.sav + +# Texpad +.texpadtmp + +# LyX +*.lyx~ + +# Kile +*.backup + +# gummi +.*.swp + +# KBibTeX +*~[0-9]* + +# TeXnicCenter +*.tps + +# auto folder when using emacs and auctex +./auto/* +*.el + +# expex forward references with \gathertags +*-tags.tex + +# standalone packages +*.sta + +# Makeindex log files +*.lpz + +# xwatermark package +*.xwm + +# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib +# option is specified. Footnotes are the stored in a file with suffix Notes.bib. +# Uncomment the next line to have this generated file ignored. +#*Notes.bib + diff --git a/generaldegreeHypersurfaces.tex b/generaldegreeHypersurfaces.tex index 431098d..0ab0f72 100644 --- a/generaldegreeHypersurfaces.tex +++ b/generaldegreeHypersurfaces.tex @@ -452,7 +452,7 @@ So we see that $\tilde{r}_2$ also takes both representatives will end to the sam We have now seen that there is a homotopy equivalence between $\CC\PP^n\setminus C_d$ and $K$. Therefore, to get a CW-complex structure on $\CC\PP^n\setminus C_d$, it suffices to get it for $K$. -We denote as $\Delta$ the $\Delta-$complex $[v_0, \dots, v_n]$ that has $n+1$ vertices, and $n+1-j$ $j$-simplices then, we can identify $S_1$ with $\Delta$, and we get a diagram: +We denote as $\Delta$ the $\Delta-$complex $[v_0, \dots, v_n]$ that has $n+1$ vertices, and $n+1-j$ $j$-simplices. Then, we can identify $S_1$ with $\Delta$, and we get a diagram: \begin{equation*} % \begin{array}{ccccc} @@ -477,24 +477,21 @@ where $\Delta^d$ is a $\Delta-$complex that we are trying to find and identify w %The $2$-faces of the simplicial complex $\Delta^d$ will then be identified to the areas between the lines $(\sqrt[d]{1-t}\xi_i, \sqrt[d]{t}\xi_j, 0)$, $(\sqrt[d]{1-t}\xi_i,0, \sqrt[d]{t}\xi_k)$ and $(0,\sqrt[d]{t}\xi_j, \sqrt[d]{1-t}\xi_k)$. First, we give a detailed description of $\Delta$. We will denote the $0$-simplices, the $i$-simplices and the $n$-simplex as: -\begin{equation} -\begin{array}{lllll} - X^j & = & v_j & = & e_j \\ - X^{(j_0,\dots,j_i)} & = & [v_{j_0},\dots,v_{j_i}]\\ - X & = & [v_0,\dots,v_n] -\end{array} -\end{equation} - +\begin{align*} + X^j & = v_j = e_j \\ + X^{(j_0,\dots,j_i)} & = [v_{j_0},\dots,v_{j_i}]\\ + X & = [v_0,\dots,v_n] +\end{align*} The face maps of $X$ and $X^{(j_0,\dots,j_i)}$ are: -\begin{equation} +\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 $k\leq i$. +\end{equation*} +where $\hat{v}$ indicates that we take out the vertex in position $k$, for $k\leq i$. an the boundary maps are given by: -\begin{equation} +\begin{equation*} \partial^i =\sum (-1)^{k}\delta^i_k \label{bordermaps} -\end{equation} +\end{equation*} We define now the $\Delta$-complex $\Delta^d$. It has $(n+1)d$ $0-$simplices, ${n+1}\choose{2}$ $d^2$ $1-$simplices, ${n+1}\choose{i}$ $d^{i+1}$ $i-$simplices and $d^{n+1}$ $n-$simplices. @@ -514,17 +511,14 @@ 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 \[ -X_{(1,\dots,\xi_i, 1, 1)}\cap\{x_j=1\forall j\neq i\}. +X_{(1,\dots,\xi_i, 1, 1)}\cap\{x_j=1,\,\forall j\neq i\}. \] -We can remark that $X_{(\xi_0, \dots, \xi_n)}=(\xi_0, \dots, \xi_n)X_{(1,\dots,1)}$, and therefore that we have a group action of $\mu^{n+1}_d$ over $S_d$. +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 -\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*} 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 \[ @@ -536,7 +530,7 @@ Then \[ C_i(\Delta^d) = \bigoplus_{(t_{j_1},\dots,t_{j_{n-i}})}\frac{R_d}{(t_{j_1} - 1,\dots,t_{j_{n-i}} - 1)} \langle X^{(k_0,\dots,k_i)}_{(1,\dots,1)}\rangle, \] -,where $t_{j_s}\neq t_{k_r}\forall s,r$ i.e, a free abelian group of rank $d^i$. Finalyy the $0$-chains are +where $t_{j_s}\neq t_{k_r}\forall s,r$ i.e, a free abelian group of rank $d^i$. Finally the $0$-chains are \[ C_0(\Delta^d) = \bigoplus_{t_j}\frac{R_d}{(t_0 - 1,\dots,t_{j-1}-1,t_{j+1}-1,\dots, t_n - 1)} \langle X^j_{(1,\dots,1)}\rangle \] @@ -551,11 +545,11 @@ a free abelian group of rank~$(n+1)d$. %\end{equation} The face maps are $R_d$-morphisms defined as -\begin{equation} +\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)} . \end{aligned} -\end{equation} +\end{equation*} %The face maps of $X_{(1,1,1)}$, $L^z_{(1,1,1)}$, $L^y_{(1,1,1)}$ and $L^x_{(1,1,1)}$ are: %\begin{equation} %\begin{array}{lll} @@ -604,7 +598,7 @@ We have then defined $\Delta^d$, the simplicial complex with which we can identi % \Delta^d_\text{proj} & \longleftrightarrow & K & \longrightarrow & \PP^2\setminus C_d \\ % \end{array} %\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 map $\Delta^d\longrightarrow\Delta^d_\text{proj}$ is given by $(x_0,\dots,x_n)\longmapsto[x_0:\dots:x_n]$. 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 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 $X^0, \dots, X^n$. @@ -614,6 +608,7 @@ The group $\mu_d^{n+1}$ also acts, but the action of the diagonal vectors is tri T_d:=\ZZ\left[\frac{\mu_d^{n+1}}{\mu_d\langle(\xi,\dots,\xi)\rangle}\right]; \] we use variables $t_0,\dots,t_n$ as for $R_d$ and we have to add the relation $t_0 \cdots t_n=1$. +This group is actualy isomorphic to $\mu_n^d$ but it is more convenient to use this presentation. We have the following chain complexes: \begin{align*} C_n(\Delta^d_\text{proj}) &= T_d\langle X_{[1:\dots:1]}\rangle,\\ @@ -631,38 +626,36 @@ 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} \begin{example}{} - We are going to compute now the simplicial complex for $S_d$ and $K$ when $d=2$, 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 $\phi(\Delta^2)=\Delta$. In this case, we have that $\mu_2=\{\xi|\xi^2=1\}=\{1,-1\}$. Then, we have that $X_{(1,1,1)}=\{(x,y,z)\in\RR^3_{\geq0} |x^2+y^2+z^2=1\}$, this is, $X_{(1,1,1)}$ is the set of points given by the positive square roots of the points in $S_1$. And the rest of $2-$simplices will be given by the action of $\mu^3_2$ on $X_{(1,1,1)}$, which will give us the $2-$simplices $X_{(1,1,-1)}$, $X_{(1,-1,1)}$, $X_{(-1,1,1)}$, $X_{(1,-1,-1)}$, $X_{(-1,1,-1)}$, $X_{(-1,-1,1)}$ and $X_{(-1,-1,-1)}$. The face maps of $X_{(1,1,1)}$ are: - \begin{equation} - \begin{array}{lllll} - \delta^2_0(X_{(1,1,1)}) & = & L^x_{(1,1,1)} & = & \{(0,\sqrt{1-t},\sqrt{t})|t\in[0,1]\}\\ - \delta^2_1(X_{(1,1,1)}) & = & L^y_{(1,1,1)} & = & \{(\sqrt{1-t},0, \sqrt{t})|t\in[0,1]\}\\ - \delta^2_2(X_{(1,1,1)}) & = & L^z_{(1,1,1)} & = & \{(\sqrt{1-t},\sqrt{t},0)|t\in[0,1]\}\\ - \end{array} - \end{equation} - + \begin{equation*} +\begin{tikzcd}[row sep=0, column sep=8pt] + \delta^2_0(X_{(1,1,1)}) \rar[equal] & L^x_{(1,1,1)} \rar[equal] & \{(0,\sqrt{1-t},\sqrt{t})|t\in[0,1]\}\\ + \delta^2_1(X_{(1,1,1)})\rar[equal]& L^y_{(1,1,1)} \rar[equal] & \{(\sqrt{1-t},0, \sqrt{t})|t\in[0,1]\}\\ + \delta^2_2(X_{(1,1,1)}) \rar[equal] & L^z_{(1,1,1)} \rar[equal] & \{(\sqrt{1-t},\sqrt{t},0)|t\in[0,1]\} +\end{tikzcd} + \end{equation*} By applying the group action as shown before we get the faces of the rest of $2-$simplices. For the $1-$simplices, the group acting would be $\mu^3_2/\mu_2$, since one of the coordinates is $0$. Therefore, the $1-$simplices will be $L^x_{(1,1,1)}$, $L^x_{(1,1,-1)}$, $L^x_{(1,-1,1)}$, $L^x_{(1,-1,-1)}$, $L^y_{(1,1,1)}$, $L^y_{(1,1,-1)}$, $L^y_{(-1,1,1)}$, $L^y_{(-1,1,-1)}$, $L^z_{(1,1,1)}$, $L^z_{(1,-1,1)}$, $L^z_{(-1,1,1)}$ and $L^z_{(-1,-1,1)}$. The face maps of $L^z_{(1,1,1)}$, $L^y_{(1,1,1)}$ and $L^x_{(1,1,1)}$ are: -\begin{equation} -\begin{array}{lllll} - \delta^1_0(L^z_{(1,1,1)}) & = & V^y_{(1,1,1)} & = & (1,0,0)\\ - \delta^1_1(L^z_{(1,1,1)}) & = & V^z_{(1,1,1)} & = & (0,1,0)\\ - \delta^1_0(L^y_{(1,1,1)}) & = & V^x_{(1,1,1)} & = & (1,0,0)\\ - \delta^1_1(L^y_{(1,1,1)}) & = & V^z_{(1,1,1)} & = & (0,0,1)\\ - \delta^1_0(L^x_{(1,1,1)}) & = & V^x_{(1,1,1)} & = & (0,1,0)\\ - \delta^1_1(L^x_{(1,1,1)}) & = & V^y_{(1,1,1)} & = & (0,0,1)\\ -\end{array} -\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 $S_2$. +\begin{equation*} +\begin{tikzcd}[row sep=0, column sep=8pt] + \delta^1_0(L^z_{(1,1,1)}) \rar[equal] & V^y_{(1,1,1)} \rar[equal] & (1,0,0)\\ + \delta^1_1(L^z_{(1,1,1)}) \rar[equal] & V^z_{(1,1,1)} \rar[equal] & (0,1,0)\\ + \delta^1_0(L^y_{(1,1,1)}) \rar[equal] & V^x_{(1,1,1)} \rar[equal] & (1,0,0)\\ + \delta^1_1(L^y_{(1,1,1)}) \rar[equal] & V^z_{(1,1,1)} \rar[equal] & (0,0,1)\\ + \delta^1_0(L^x_{(1,1,1)}) \rar[equal] & V^x_{(1,1,1)} \rar[equal] & (0,1,0)\\ + \delta^1_1(L^x_{(1,1,1)}) \rar[equal] & V^y_{(1,1,1)} \rar[equal] & (0,0,1)\\ +\end{tikzcd} +\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. %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{array}{rll}