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+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\usepackage{tikz}
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Un 
+\series bold
+plano afín
+\series default
+ es una terna 
+\begin_inset Formula $\mathbb{A}=({\cal P},{\cal L},\epsilon)$
+\end_inset
+
+ formada por los conjuntos 
+\begin_inset Formula ${\cal P},{\cal L}\neq\emptyset$
+\end_inset
+
+, cuyos elementos se llaman 
+\series bold
+puntos
+\series default
+ y 
+\series bold
+rectas
+\series default
+, respectivamente, y la 
+\series bold
+relación de incidencia
+\series default
+ 
+\begin_inset Formula $\epsilon\subseteq{\cal P}\times{\cal L}$
+\end_inset
+
+, que satisface que 
+\begin_inset Formula $\forall P,Q\in{\cal P},\ell\in{\cal L}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+
+\family sans
+\begin_inset Formula $P\neq Q\implies\exists!\ell\in{\cal L}:P,Q\epsilon\ell$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\exists P,Q,R\in{\cal P}:\nexists\ell\in{\cal L}:P,Q,R\epsilon\ell$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $P\not\epsilon\ell\implies\exists!m\in{\cal L}:(P\epsilon m\land\nexists Q\in{\cal P}:Q\epsilon\ell,m)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $P\in{\cal P}$
+\end_inset
+
+ es 
+\series bold
+incidente
+\series default
+ con 
+\begin_inset Formula $\ell\in{\cal L}$
+\end_inset
+
+ si 
+\begin_inset Formula $P\epsilon\ell$
+\end_inset
+
+, y 
+\begin_inset Formula $\ell,m\in{\cal L}$
+\end_inset
+
+ son 
+\series bold
+paralelas
+\series default
+ si 
+\begin_inset Formula $\nexists P:P\epsilon\ell,m$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $V$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio vectorial y 
+\begin_inset Formula $\dim_{\mathbb{K}}V\geq2$
+\end_inset
+
+, definimos el plano afín 
+\begin_inset Formula $\mathbb{A}(V):=({\cal P}(V),{\cal L}(V),\in)$
+\end_inset
+
+ con 
+\begin_inset Formula ${\cal P}(V):=V$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal L}(V)=\{\vec{v}+<\vec{w}>\}_{\vec{v},\vec{w}\in V,\vec{w}\neq0}$
+\end_inset
+
+, y escribimos 
+\begin_inset Formula $\mathbb{A}^{n}(\mathbb{K}):=\mathbb{A}(\mathbb{K}^{n})$
+\end_inset
+
+.
+ Llamamos a 
+\begin_inset Formula $\mathbb{A}^{2}(\mathbb{R})$
+\end_inset
+
+ el 
+\series bold
+plano afín usual
+\series default
+.
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+plano proyectivo
+\series default
+ es una terna 
+\begin_inset Formula $\mathbb{P}=({\cal P},{\cal L},\epsilon)$
+\end_inset
+
+ similar a un plano afín pero cambiando el último axioma por que 
+\begin_inset Formula $\forall\ell,m\in{\cal L}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+3.
+\end_layout
+
+\end_inset
+
+
+\family sans
+
+\begin_inset Formula $\exists P\in{\cal P}:P\epsilon\ell,m$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+4.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $\exists P,Q,R\in{\cal P}:(P\neq Q\neq R\neq P\land P,Q,R\in\ell)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+principio de dualidad para planos proyectivos
+\series default
+ afirma que si 
+\begin_inset Formula $\pi:=({\cal P},{\cal L},\epsilon)$
+\end_inset
+
+ es un plano proyectivo entonces 
+\begin_inset Formula $\pi^{*}:=({\cal L},{\cal P},\epsilon^{*})$
+\end_inset
+
+ con 
+\begin_inset Formula $\ell\epsilon^{*}P\iff P\epsilon\ell$
+\end_inset
+
+ también lo es.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall\ell,m\in{\cal L},(\ell\neq m\implies\exists!P\in{\cal P}:P\epsilon\ell,m)$
+\end_inset
+
+: El axioma 3 asegura que 
+\begin_inset Formula $P$
+\end_inset
+
+ existe.
+ Ahora bien, si existiera otro 
+\begin_inset Formula $Q\neq P$
+\end_inset
+
+ con 
+\begin_inset Formula $Q\epsilon\ell,m$
+\end_inset
+
+, por el axioma 1 se tendría 
+\begin_inset Formula $\ell=m\#$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\exists\ell,m,n\in{\cal L}:\nexists P\in{\cal P}:P\epsilon\ell,m,n$
+\end_inset
+
+: El axioma 2 nos dice que hay 3 puntos 
+\begin_inset Formula $Q,R,S\in{\cal P}$
+\end_inset
+
+ para los que 
+\begin_inset Formula $\nexists\ell\in{\cal L}:Q,R,S\epsilon\ell$
+\end_inset
+
+.
+ Si fueran 
+\begin_inset Formula $Q=R\neq S$
+\end_inset
+
+, el axioma 1 nos dice que existe una recta que los contiene, y si fueran
+ 
+\begin_inset Formula $Q=R=S$
+\end_inset
+
+, podríamos tomar uno de los puntos del axioma 4 (para alguna recta) como
+ punto distinto a este para aplicar el axioma 1.
+ Por tanto los 3 puntos son distintos.
+ Sean ahora 
+\begin_inset Formula $\ell:=QR$
+\end_inset
+
+, 
+\begin_inset Formula $m:=RS$
+\end_inset
+
+ y 
+\begin_inset Formula $n:=SQ$
+\end_inset
+
+ (aplicando el axioma 1).
+ Si hubiera un punto 
+\begin_inset Formula $P\epsilon\ell,m,n$
+\end_inset
+
+ (podemos suponer 
+\begin_inset Formula $P\neq Q,R$
+\end_inset
+
+), entonces por el axioma 1 
+\begin_inset Formula $n=PQ=\ell=PR=m$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $Q,R,S\in\ell\#$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall P,Q\in{\cal P},\exists\ell\in{\cal L}:P,Q\epsilon\ell$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $P\neq Q$
+\end_inset
+
+, esto nos lo asegura el axioma 1.
+ Para poder aplicarlo con 
+\begin_inset Formula $P=Q$
+\end_inset
+
+, tomamos un punto de los dados por el axioma 4 que sea distinto a 
+\begin_inset Formula $P$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall P\in{\cal P},\exists\ell,m,n\in{\cal L}:(\ell\neq m\neq n\neq\ell\land P\epsilon\ell,m,n)$
+\end_inset
+
+.
+ Tomamos los puntos 
+\begin_inset Formula $Q,R,S$
+\end_inset
+
+ dados por el axioma 2, que ya hemos visto que deben ser distintos.
+ Podemos suponer 
+\begin_inset Formula $P\neq Q,R$
+\end_inset
+
+, y entonces podemos suponer 
+\begin_inset Formula $PQ\neq PR$
+\end_inset
+
+.
+ En efecto, si fueran iguales sería 
+\begin_inset Formula $P\epsilon QR$
+\end_inset
+
+ y 
+\begin_inset Formula $S\not\epsilon QR=PR$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $P\neq S$
+\end_inset
+
+ y además 
+\begin_inset Formula $PS\neq PR$
+\end_inset
+
+, y podríamos tomar 
+\begin_inset Formula $S$
+\end_inset
+
+ en vez de 
+\begin_inset Formula $R$
+\end_inset
+
+.
+ Ahora tomamos 
+\begin_inset Formula $QR$
+\end_inset
+
+ que, por el axioma 4, contiene un tercer punto 
+\begin_inset Formula $T\neq Q,R$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $P\neq T$
+\end_inset
+
+ (si fuera 
+\begin_inset Formula $P=T$
+\end_inset
+
+ se tendría 
+\begin_inset Formula $PQ=PR\#$
+\end_inset
+
+) y 
+\begin_inset Formula $PT\neq PQ,PR$
+\end_inset
+
+ (si fuera, por ejemplo, 
+\begin_inset Formula $PT=PQ$
+\end_inset
+
+, se tendría 
+\begin_inset Formula $PQ=TQ=TR=QR\#$
+\end_inset
+
+).
+ Por tanto, 
+\begin_inset Formula $\ell:=PQ$
+\end_inset
+
+, 
+\begin_inset Formula $m:=PT$
+\end_inset
+
+ y 
+\begin_inset Formula $n:=PR$
+\end_inset
+
+ cumplen las condiciones.
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula $\pi=({\cal P},{\cal L},\epsilon)$
+\end_inset
+
+ y 
+\begin_inset Formula $\pi'=({\cal P}',{\cal L}',\epsilon')$
+\end_inset
+
+ dos planos proyectivos, un 
+\series bold
+isomorfismo
+\series default
+ de 
+\begin_inset Formula $\pi$
+\end_inset
+
+ a 
+\begin_inset Formula $\pi'$
+\end_inset
+
+ es un par 
+\begin_inset Formula $(f:{\cal P}\rightarrow{\cal P}',f':{\cal L}\rightarrow{\cal L}')$
+\end_inset
+
+ de biyecciones tal que 
+\begin_inset Formula $\forall P\in{\cal P},\ell\in{\cal L},(P\epsilon\ell\implies f(P)\epsilon'f'(\ell))$
+\end_inset
+
+.
+ Si existe, decimos que 
+\begin_inset Formula $\pi$
+\end_inset
+
+ y 
+\begin_inset Formula $\pi'$
+\end_inset
+
+ son 
+\series bold
+isomorfos
+\series default
+, si y sólo si existe una biyección 
+\begin_inset Formula $f:{\cal P}\rightarrow{\cal P}'$
+\end_inset
+
+ que lleva ternas de puntos alineados a ternas de puntos alineados.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Obvio.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Dada una recta 
+\begin_inset Formula $\ell$
+\end_inset
+
+, por el axioma 4 existen tres puntos 
+\begin_inset Formula $P,Q,R$
+\end_inset
+
+ distintos sobre la recta.
+ Definimos 
+\begin_inset Formula $f':{\cal L}\rightarrow{\cal L}'$
+\end_inset
+
+ tal que 
+\begin_inset Formula $f'(\ell):=\overline{f(P)f(Q)}$
+\end_inset
+
+.
+ Para ver que está bien definida, sean 
+\begin_inset Formula $P',Q'\epsilon\ell$
+\end_inset
+
+, 
+\begin_inset Formula $P'\neq Q'$
+\end_inset
+
+ con 
+\begin_inset Formula $\{P,Q\}\neq\{P',Q'\}$
+\end_inset
+
+ (podemos suponer 
+\begin_inset Formula $P\neq P',Q'$
+\end_inset
+
+ y 
+\begin_inset Formula $P'\neq P,Q$
+\end_inset
+
+).
+ Entonces 
+\begin_inset Formula $f'(\ell)=\overline{f(P')f(Q')}$
+\end_inset
+
+, pero como 
+\begin_inset Formula $P,Q,P',Q'$
+\end_inset
+
+ están alineados, 
+\begin_inset Formula $f(P),f(Q),f(P'),f(Q')$
+\end_inset
+
+ también lo están, y 
+\begin_inset Formula $\overline{f(P')f(Q')}=\overline{f(P)f(Q)}$
+\end_inset
+
+.
+ Sean 
+\begin_inset Formula $\ell:=\overline{PQ}$
+\end_inset
+
+ y 
+\begin_inset Formula $R\epsilon\ell$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f'(\ell)=\overline{f(P)f(Q)}$
+\end_inset
+
+, pero como 
+\begin_inset Formula $P,Q,R$
+\end_inset
+
+ están alineados, 
+\begin_inset Formula $f(R)\epsilon'\overline{f(P)f(Q)}=f'(\ell)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Construcción de 
+\begin_inset Formula $\mathbb{P}^{2}(\mathbb{K})$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si en el espacio afín 
+\begin_inset Formula $\mathbb{A}:=\mathbb{A}(W)$
+\end_inset
+
+ para cierto espacio vectorial 
+\begin_inset Formula $W$
+\end_inset
+
+ definimos la relación de equivalencia 
+\begin_inset Formula $\ell\sim\ell':\iff\ell\parallel\ell'$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\overline{\mathbb{A}}:=({\cal P}',{\cal L}',\in)$
+\end_inset
+
+ con 
+\begin_inset Formula ${\cal P}':={\cal P}\cup({\cal L}/\sim)$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal L}':=\{\ell\cup\{[\ell]\}\}_{\ell\in{\cal L}}\cup\{{\cal L}/\sim\}$
+\end_inset
+
+ es un plano proyectivo al que llamamos 
+\series bold
+extensión proyectiva
+\series default
+ de 
+\begin_inset Formula $\mathbb{A}$
+\end_inset
+
+.
+ Llamamos 
+\series bold
+puntos afines
+\series default
+ a los de 
+\begin_inset Formula ${\cal P}$
+\end_inset
+
+ y 
+\series bold
+puntos del infinito
+\series default
+ a los de 
+\begin_inset Formula ${\cal L}/\sim$
+\end_inset
+
+.
+ De igual modo, llamamos 
+\series bold
+rectas extendidas
+\series default
+ a las 
+\begin_inset Formula $\overline{\ell}:=\ell\cup\{[\ell]\}$
+\end_inset
+
+ y 
+\series bold
+recta del infinito
+\series default
+ a 
+\begin_inset Formula $\ell_{\infty}:={\cal L}/\sim$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dado el 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio vectorial 
+\begin_inset Formula $W\equiv\mathbb{K}^{3}$
+\end_inset
+
+, si 
+\begin_inset Formula ${\cal P}(W):=\{\text{rectas vectoriales de }W\}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal L}(W):=\{\text{planos vectoriales de }W\}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $({\cal P}(W),{\cal L}(W),\subseteq)$
+\end_inset
+
+ es un plano proyectivo.
+ Llamamos 
+\series bold
+plano proyectivo en 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+
+\series default
+ a 
+\begin_inset Formula $\mathbb{P}^{2}(\mathbb{K}):=({\cal P}(\mathbb{K}^{3}),{\cal L}(\mathbb{K}^{3}),\subseteq)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $<\vec{v}>\neq<\vec{w}>\implies\exists!\pi\in{\cal L}(W):<\vec{v}>,<\vec{w}>\subseteq\pi$
+\end_inset
+
+: 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}$
+\end_inset
+
+ son LI, luego necesariamente 
+\begin_inset Formula $\pi=<\vec{v},\vec{w}>$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\exists\vec{u},\vec{v},\vec{w}\in W:\nexists\pi\in{\cal L}(W):<\vec{u},\vec{v},\vec{w}>\subseteq\pi$
+\end_inset
+
+: Basta tomar una base de 
+\begin_inset Formula $W$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\exists\vec{u}\in W:<\vec{u}>\subseteq\pi$
+\end_inset
+
+: Si 
+\begin_inset Formula $\pi=<\vec{v},\vec{w}>$
+\end_inset
+
+, basta tomar 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\exists\vec{u},\vec{v},\vec{w}\in W:(<\vec{u}>\neq<\vec{v}>\neq<\vec{w}>\neq<\vec{u}>\land<\vec{u},\vec{v},\vec{w}>\in\pi)$
+\end_inset
+
+: Si 
+\begin_inset Formula $\pi=<\vec{v},\vec{w}>$
+\end_inset
+
+, basta tomar 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+, 
+\begin_inset Formula $\vec{w}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{v}+\vec{w}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{sloppypar}
+\end_layout
+
+\end_inset
+
+Dado un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio vectorial 
+\begin_inset Formula $W$
+\end_inset
+
+ de dimensión 2, 
+\begin_inset Formula $\overline{\mathbb{A}(W)}$
+\end_inset
+
+ es isomorfo a 
+\begin_inset Formula $\mathbb{P}(W\times\mathbb{K})$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula ${\cal P}:=W\cup\{[\ell]\}_{\ell\text{ recta afín de }W}$
+\end_inset
+
+ el conjunto de puntos de 
+\begin_inset Formula $\overline{\mathbb{A}(W)}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal P}':=\{\text{rectas vectoriales de }W\times\mathbb{K}\}$
+\end_inset
+
+ el conjunto de puntos de 
+\begin_inset Formula $\mathbb{P}(W\times\mathbb{K})$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $\sigma:{\cal P}\rightarrow{\cal P}'$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\sigma(u)=<(u,1)>\forall u\in W$
+\end_inset
+
+ y 
+\begin_inset Formula $\sigma([<u>])=<(u,0)>\forall u\in W$
+\end_inset
+
+, una biyección cuya inversa viene dada por 
+\begin_inset Formula $\sigma^{-1}(<(u,0)>)=[<(u,0)>]\forall u\in W$
+\end_inset
+
+ y 
+\begin_inset Formula $\sigma^{-1}(<(u,\lambda)>)=\frac{u}{\lambda}\forall u\in W,\lambda\neq0$
+\end_inset
+
+.
+ Veamos que lleva ternas de puntos alineados a ternas de puntos alineados:
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{sloppypar}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si los tres puntos son afines y suponemos 
+\begin_inset Formula $u_{1}\neq0,u_{2}$
+\end_inset
+
+, que estén alineados significa que 
+\begin_inset Formula $u_{2}=\lambda u_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $u_{3}=\mu u_{1}$
+\end_inset
+
+ para 
+\begin_inset Formula $\lambda\neq1$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\sigma(u_{1})=<(u_{1},1)>$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma(u_{2})=<(\lambda u_{1},1)>$
+\end_inset
+
+ y 
+\begin_inset Formula $\sigma(u_{3})=<(\mu u_{1},1)>$
+\end_inset
+
+, pero 
+\begin_inset Formula $\frac{\lambda-\mu}{\lambda-1}(u_{1},1)+\frac{\mu-1}{\lambda-1}(\lambda u_{1},1)=(\mu u_{1},1)$
+\end_inset
+
+, luego las tres rectas se encuentran en un plano.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $u_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $u_{2}$
+\end_inset
+
+ son afines y 
+\begin_inset Formula $[<u_{3}>]$
+\end_inset
+
+ es del infinito, que estén alineados significa que 
+\begin_inset Formula $u_{2}=u_{1}+\lambda u_{3}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\sigma(u_{1})=<(u_{1},1)>$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma(u_{2})=<(u_{1}+\lambda u_{3},1)>$
+\end_inset
+
+ y 
+\begin_inset Formula $\sigma(u_{3})=<(u_{3},0)>$
+\end_inset
+
+.
+ Pero 
+\begin_inset Formula $(u_{1},1)+\lambda(u_{3},0)=(u_{1}+\lambda_{3},1)$
+\end_inset
+
+, luego las tres rectas están en el mismo plano.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $u_{1}$
+\end_inset
+
+ es afín y 
+\begin_inset Formula $[<u_{2}>],[<u_{3}>]$
+\end_inset
+
+ son del infinito, que estén alineados significa que 
+\begin_inset Formula $u_{2}=u_{3}$
+\end_inset
+
+, y entonces es claro que hay una recta que une 
+\begin_inset Formula $\sigma(u_{1})$
+\end_inset
+
+ con 
+\begin_inset Formula $\sigma([<u_{2}=u_{3}>])$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si los tres puntos son del infinito, siempre están alineados, pero entonces
+ para 
+\begin_inset Formula $i\in\{1,2,3\}$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma([<u_{i}>])=<(u_{i},0)>\in W\times\{0\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Referencias proyectivas
+\end_layout
+
+\begin_layout Standard
+Tres puntos 
+\begin_inset Formula $P,Q,R\in\mathbb{P}^{2}(\mathbb{K})$
+\end_inset
+
+ son (
+\series bold
+proyectivamente
+\series default
+)
+\series bold
+ independientes
+\series default
+ si los vectores que los representan forman una base de 
+\begin_inset Formula $\mathbb{K}^{3}$
+\end_inset
+
+.
+ Una 
+\series bold
+referencia proyectiva
+\series default
+ o 
+\series bold
+referencial proyectivo
+\series default
+ en 
+\begin_inset Formula $\mathbb{P}^{2}(\mathbb{K})$
+\end_inset
+
+ es una cuaterna 
+\begin_inset Formula ${\cal R}:=(P,Q,R,U)$
+\end_inset
+
+ de puntos tales que tres puntos cualesquiera de ellos son independientes.
+\end_layout
+
+\begin_layout Standard
+Todo referencial proyectivo de 
+\begin_inset Formula $\mathbb{P}^{2}(\mathbb{K})$
+\end_inset
+
+ admite una base 
+\begin_inset Formula ${\cal B}:=(v_{1},v_{2},v_{3})$
+\end_inset
+
+ de 
+\begin_inset Formula $\mathbb{K}^{3}$
+\end_inset
+
+ tal que
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $P=<v_{1}>$
+\end_inset
+
+, 
+\begin_inset Formula $Q=<v_{2}>$
+\end_inset
+
+, 
+\begin_inset Formula $R=<v_{3}>$
+\end_inset
+
+ y 
+\begin_inset Formula $U=<v_{1}+v_{2}+v_{3}>$
+\end_inset
+
+, única salvo multiplicación simultánea de los 3 vectores por un escalar
+ no nulo.
+ A esta base la llamamos 
+\series bold
+base asociada
+\series default
+ al referencial 
+\begin_inset Formula ${\cal R}$
+\end_inset
+
+, y el punto 
+\begin_inset Formula $U$
+\end_inset
+
+ es el 
+\series bold
+punto unidad
+\series default
+ del referencial.
+ 
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula $<v_{1}>,<v_{2}>,<v_{3}>$
+\end_inset
+
+ son no alineados en 
+\begin_inset Formula $\mathbb{P}^{2}(\mathbb{K})$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $(v_{1},v_{2},v_{3})$
+\end_inset
+
+ es una base.
+ Entonces, si 
+\begin_inset Formula $P=:<u_{1}>$
+\end_inset
+
+, 
+\begin_inset Formula $Q=:<u_{2}>$
+\end_inset
+
+ y 
+\begin_inset Formula $R=:<u_{3}>$
+\end_inset
+
+, podemos escribir 
+\begin_inset Formula $U=:<u>$
+\end_inset
+
+ con 
+\begin_inset Formula $u:=\alpha_{1}u_{1}+\alpha_{2}u_{2}+\alpha_{3}u_{3}$
+\end_inset
+
+ con 
+\begin_inset Formula $(\alpha_{1},\alpha_{2},\alpha_{3})\neq(0,0,0)$
+\end_inset
+
+.
+ Entonces hacemos 
+\begin_inset Formula $v_{i}:=\alpha_{i}u_{i}$
+\end_inset
+
+ para 
+\begin_inset Formula $i\in\{1,2,3\}$
+\end_inset
+
+, y sabemos que 
+\begin_inset Formula $\alpha_{1},\alpha_{2},\alpha_{3}\neq0$
+\end_inset
+
+, pues si fuera algún 
+\begin_inset Formula $\alpha_{i}=0$
+\end_inset
+
+, 
+\begin_inset Formula $u$
+\end_inset
+
+ sería linealmente dependiente con 
+\begin_inset Formula $u_{j}$
+\end_inset
+
+ y 
+\begin_inset Formula $u_{k}$
+\end_inset
+
+ para 
+\begin_inset Formula $j,k\neq i$
+\end_inset
+
+ y serían alineados, luego 
+\begin_inset Formula $(v_{1},v_{2},v_{3})$
+\end_inset
+
+ es una base que satisface las condiciones.
+ Ahora bien, si existe 
+\begin_inset Formula ${\cal B}'=(v'_{1},v'_{2},v'_{3})$
+\end_inset
+
+ que también satisface las condiciones, necesariamente 
+\begin_inset Formula $<v'_{1}>=P=<v_{1}>$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $v'_{1}=\lambda_{1}v_{1}$
+\end_inset
+
+ para algún 
+\begin_inset Formula $\lambda_{1}\neq0$
+\end_inset
+
+, y lo mismo sucede con 
+\begin_inset Formula $v'_{2}$
+\end_inset
+
+ y 
+\begin_inset Formula $v'_{3}$
+\end_inset
+
+, pero entonces 
+\begin_inset Formula $<v'_{1}+v'_{2}+v'_{3}>=<\lambda_{1}v'_{1}+\lambda_{2}v'_{2}+\lambda_{3}v'_{3}>=U=<v_{1}+v_{2}+v_{3}>$
+\end_inset
+
+, y es claro que 
+\begin_inset Formula $\lambda_{1}=\lambda_{2}=\lambda_{3}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dado 
+\begin_inset Formula $P\in\mathbb{P}^{2}(\mathbb{K})$
+\end_inset
+
+, decimos que sus 
+\series bold
+coordenadas homogéneas
+\series default
+ respecto a la base 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ a
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+so
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+cia
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+da al referencial 
+\begin_inset Formula ${\cal R}$
+\end_inset
+
+ son 
+\begin_inset Formula $x,y,z$
+\end_inset
+
+ (
+\begin_inset Formula $P:=[x,y,z]$
+\end_inset
+
+) si 
+\begin_inset Formula $P=:<\vec{u}>$
+\end_inset
+
+ con 
+\begin_inset Formula $[\vec{u}]_{{\cal B}}=(x,y,z)$
+\end_inset
+
+.
+ Estas son únicas salvo multiplicación de las tres por un escalar no nulo.
+ Tres puntos de coordenadas homogéneas 
+\begin_inset Formula $[a,b,c]$
+\end_inset
+
+, 
+\begin_inset Formula $[d,e,f]$
+\end_inset
+
+ y 
+\begin_inset Formula $[g,h,i]$
+\end_inset
+
+ son proyectivamente independientes si y sólo si
+\begin_inset Formula 
+\[
+\left|\begin{array}{ccc}
+a & b & c\\
+d & e & f\\
+g & h & i
+\end{array}\right|\neq0
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\begin_inset Formula $[a,b,c]^{*}$
+\end_inset
+
+ a la recta en 
+\begin_inset Formula $\mathbb{P}^{2}(\mathbb{K})$
+\end_inset
+
+ dada por 
+\begin_inset Formula $ax+by+cz=0$
+\end_inset
+
+.
+ Las rectas 
+\begin_inset Formula $\ell:=[a_{1},b_{1},c_{1}]^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $m:=[a_{2},b_{2},c_{2}]^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula $n:=[a_{3},b_{3},c_{3}]^{*}$
+\end_inset
+
+ son 
+\series bold
+congruentes
+\series default
+ (se cortan) si y sólo si
+\begin_inset Formula 
+\[
+\left|\begin{array}{ccc}
+a_{1} & b_{1} & c_{1}\\
+a_{2} & b_{2} & c_{2}\\
+a_{3} & b_{3} & c_{3}
+\end{array}\right|=0
+\]
+
+\end_inset
+
+
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula 
+\begin{multline*}
+\exists P\in\ell,m,n\iff\exists(x_{0},y_{0},x_{0})\neq0:\forall i\in\{1,2,3\},a_{i}x_{0}+b_{i}y_{0}+c_{i}z_{0}=0\iff\\
+\iff\dim\left\{ \left(\begin{array}{ccc}
+a_{1} & b_{1} & c_{1}\\
+a_{2} & b_{2} & c_{2}\\
+a_{3} & b_{3} & c_{3}
+\end{array}\right)\left(\begin{array}{c}
+x\\
+y\\
+z
+\end{array}\right)=\left(\begin{array}{c}
+0\\
+0\\
+0
+\end{array}\right)\right\} >0
+\end{multline*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Teoremas de Desargues y Pappus
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+teorema de Desargues
+\series default
+ afirma que, dados dos triángulos 
+\begin_inset Formula $ABC$
+\end_inset
+
+ y 
+\begin_inset Formula $A'B'C'$
+\end_inset
+
+ sin vértices ni lados comunes, si las rectas que unen vértices correspondientes
+ (
+\begin_inset Formula $AA'$
+\end_inset
+
+, 
+\begin_inset Formula $BB'$
+\end_inset
+
+ y 
+\begin_inset Formula $CC'$
+\end_inset
+
+) se cortan en un punto, los puntos de corte de lados correspondientes están
+ alineados.
+ Un plano proyectivo es 
+\series bold
+desarguesiano
+\series default
+ si satisface este teorema.
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+teorema de Pappus
+\series default
+ afirma que, dados tres puntos distintos 
+\begin_inset Formula $A,B,C$
+\end_inset
+
+ en una recta y 
+\begin_inset Formula $A',B',C'$
+\end_inset
+
+ en otra, los puntos 
+\begin_inset Formula $L\in AB'\cap A'B$
+\end_inset
+
+, 
+\begin_inset Formula $M\in AC'\cap A'C$
+\end_inset
+
+ y 
+\begin_inset Formula $N\in BC'\cap B'C$
+\end_inset
+
+ están alineados.
+ Un plano proyectivo es 
+\series bold
+papiano
+\series default
+ si satisface este teorema.
+\end_layout
+
+\begin_layout Standard
+Un plano proyectivo 
+\begin_inset Formula $\pi$
+\end_inset
+
+ es papiano y desarguesiano si y sólo si es isomorfo a 
+\begin_inset Formula $\mathbb{P}(V)$
+\end_inset
+
+ para algún 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio vectorial tridimensional, si y sólo si es isomorfo a 
+\begin_inset Formula $\mathbb{P}^{2}(\mathbb{K})$
+\end_inset
+
+.
+ En tal caso, el cuerpo 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+ de las dos últimas condiciones es el mismo y está unívocamente determinado
+ por 
+\begin_inset Formula $\pi$
+\end_inset
+
+ salvo isomorfismo.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $2\iff3]$
+\end_inset
+
+ Sea 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ una base del 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio tridimensional 
+\begin_inset Formula $V$
+\end_inset
+
+, 
+\begin_inset Formula $[\cdot]_{{\cal B}}:V\longrightarrow\mathbb{K}^{3}$
+\end_inset
+
+ define un isomorfismo entre los puntos de 
+\begin_inset Formula $\mathbb{P}(V)$
+\end_inset
+
+ y los de 
+\begin_inset Formula $\mathbb{P}(\mathbb{K}^{3})=\mathbb{P}^{2}(\mathbb{K})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $3\implies1]$
+\end_inset
+
+ Probemos el teorema de Desargues.
+ Sean 
+\begin_inset Formula $O:=[\vec{o}]$
+\end_inset
+
+ el punto de corte entre las tres rectas, 
+\begin_inset Formula $A:=[\vec{a}]$
+\end_inset
+
+, 
+\begin_inset Formula $B:=[\vec{b}]$
+\end_inset
+
+ y 
+\begin_inset Formula $C:=[\vec{c}]$
+\end_inset
+
+ con 
+\begin_inset Formula $\vec{a},\vec{b},\vec{c},\vec{o}\neq0$
+\end_inset
+
+, como 
+\begin_inset Formula $O,A,A'$
+\end_inset
+
+ están alineados, debe ser 
+\begin_inset Formula $A'=[\lambda\vec{o}+\mu\vec{a}]$
+\end_inset
+
+ con 
+\begin_inset Formula $\lambda\neq0$
+\end_inset
+
+ (si fuera 
+\begin_inset Formula $\lambda=0$
+\end_inset
+
+ sería 
+\begin_inset Formula $A=A'$
+\end_inset
+
+ y 
+\begin_inset Formula $AA'$
+\end_inset
+
+ no tendría sentido) y, dividiendo por 
+\begin_inset Formula $\lambda$
+\end_inset
+
+, 
+\begin_inset Formula $A'=:[\vec{o}+\alpha\vec{a}]$
+\end_inset
+
+.
+ Análogamente 
+\begin_inset Formula $B'=:[\vec{o}+\beta\vec{b}]$
+\end_inset
+
+ y 
+\begin_inset Formula $C'=:[\vec{o}+\gamma\vec{c}]$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $\alpha\vec{a}-\beta\vec{b}=(\vec{o}+\alpha\vec{a})-(\vec{o}+\beta\vec{b})$
+\end_inset
+
+, tenemos que 
+\begin_inset Formula $AB\cap A'B'=\{[\alpha\vec{a}-\beta\vec{b}]\}$
+\end_inset
+
+, y del mismo modo 
+\begin_inset Formula $AC\cap A'C'=\{[\alpha\vec{a}-\gamma\vec{c}]\}$
+\end_inset
+
+ y 
+\begin_inset Formula $BC\cap B'C'=\{[\beta\vec{b}-\gamma\vec{c}]\}$
+\end_inset
+
+.
+ Estos tres puntos están alineados, pues 
+\begin_inset Formula $(\alpha\vec{a}-\beta\vec{b})-(\alpha\vec{a}-\gamma\vec{c})+(\beta\vec{b}-\gamma\vec{c})=0$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Para el teorema de Pappus, consideremos la referencia proyectiva 
+\begin_inset Formula ${\cal R}:=(A',A,B,B')$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $A'=[1,0,0]$
+\end_inset
+
+, 
+\begin_inset Formula $A=[0,1,0]$
+\end_inset
+
+, 
+\begin_inset Formula $B=[0,0,1]$
+\end_inset
+
+ y 
+\begin_inset Formula $B'=[1,1,1]$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $C\epsilon AB$
+\end_inset
+
+, debe ser 
+\begin_inset Formula $C=[0,\alpha,\beta]$
+\end_inset
+
+ con 
+\begin_inset Formula $\alpha,\beta\neq0$
+\end_inset
+
+, luego 
+\begin_inset Formula $C=[0,1,c]$
+\end_inset
+
+ para algún 
+\begin_inset Formula $c\neq0$
+\end_inset
+
+.
+ De forma parecida, 
+\begin_inset Formula $C'=[c',1,1]$
+\end_inset
+
+.
+ Entonces
+\begin_inset Formula 
+\begin{eqnarray*}
+AB':x=z & AC':x=c'z & BC':x=c'y\\
+A'B:y=0 & A'C:z=cy & B'C:(c-1)x-cy+z=0
+\end{eqnarray*}
+
+\end_inset
+
+de donde 
+\begin_inset Formula $AB'\cap A'B=\{[1,0,1]\}$
+\end_inset
+
+, 
+\begin_inset Formula $AC'\cap A'C=\{[cc',1,c]\}$
+\end_inset
+
+ y 
+\begin_inset Formula $BC'\cap B'C=\{[c',1,c+c'-cc']\}$
+\end_inset
+
+, y los tres puntos están alineados porque
+\begin_inset Formula 
+\[
+\left|\begin{array}{ccc}
+1 & 0 & 1\\
+cc' & 1 & c\\
+c' & 1 & c+c'-cc'
+\end{array}\right|=0
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Ampliación proyectiva
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\begin_inset Formula $\mathbb{K}[x_{1},\dots,x_{n}]$
+\end_inset
+
+ al conjunto de polinomios de 
+\begin_inset Formula $n$
+\end_inset
+
+ variables sobre 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+, y decimos que 
+\begin_inset Formula $F\in\mathbb{K}[x_{1},\dots,x_{n}]$
+\end_inset
+
+ es 
+\series bold
+homogéneo
+\series default
+ si todos sus monomios tienen el mismo grado.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{sloppypar}
+\end_layout
+
+\end_inset
+
+Dado 
+\begin_inset Formula $f\in\mathbb{K}[x_{1},\dots,x_{n}]$
+\end_inset
+
+, su 
+\series bold
+homogeneización
+\series default
+ es el polinomio homogéneo 
+\begin_inset Formula $f^{*}\in\mathbb{K}[x_{1},\dots,x_{n+1}]$
+\end_inset
+
+ dado por 
+\begin_inset Formula $f^{*}(x_{1},\dots,x_{n+1})=x_{n+1}^{d}f(\frac{x_{1}}{x_{n+1}},\dots,\frac{x_{n}}{x_{n+1}})$
+\end_inset
+
+, siendo 
+\begin_inset Formula $d$
+\end_inset
+
+ el grado de 
+\begin_inset Formula $f$
+\end_inset
+
+, es decir, el máximo de los grados de sus monomios.
+ La 
+\series bold
+deshomogeneización
+\series default
+ de 
+\begin_inset Formula $F\in\mathbb{K}[x_{1},\dots,x_{n+1}]$
+\end_inset
+
+ es 
+\begin_inset Formula $F_{*}\in\mathbb{K}[x_{1},\dots,x_{n}]$
+\end_inset
+
+ dado por 
+\begin_inset Formula $F_{*}(x_{1},\dots,x_{n})=F(x_{1},\dots,x_{n},1)$
+\end_inset
+
+.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{sloppypar}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Dado 
+\begin_inset Formula $f\in\mathbb{K}[x_{1},\dots,x_{n}]$
+\end_inset
+
+, 
+\begin_inset Formula $(f^{*})_{*}=f$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Si 
+\begin_inset Formula $f(x_{1},\dots,x_{n}):=\sum_{i=1}^{k}\prod_{j=1}^{d_{i}}x_{a_{ij}}$
+\end_inset
+
+, entonces 
+\begin_inset Formula 
+\[
+f^{*}(x_{1},\dots,x_{n+1})=\sum_{i=1}^{k}x_{n+1}^{\max\{d_{i}\}}\prod_{j=1}^{d_{i}}\frac{x_{a_{ij}}}{x_{n+1}}=\sum_{i=1}^{k}x_{n+1}^{\max\{d_{i}\}-d_{i}}\prod_{j=1}^{d_{i}}x_{a_{ij}}
+\]
+
+\end_inset
+
+ y 
+\begin_inset Formula $(f^{*})_{*}(x_{1},\dots,x_{n})=\sum_{i=1}^{k}\prod_{j=1}^{d_{i}}x_{a_{ij}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Dado 
+\begin_inset Formula $F\in\mathbb{K}[x_{1},\dots,x_{n+1}]$
+\end_inset
+
+ homogéneo, 
+\begin_inset Formula $F=x_{n+1}^{k}(F_{*})^{*}$
+\end_inset
+
+, siendo 
+\begin_inset Formula $k$
+\end_inset
+
+ la mayor potencia de 
+\begin_inset Formula $x_{n+1}$
+\end_inset
+
+ que divide a todos los monomios de 
+\begin_inset Formula $F$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Si 
+\begin_inset Formula $F(x_{1},\dots,x_{n+1}):=\sum_{i=1}^{k}x_{n+1}^{b_{i}}\prod_{j=1}^{d-b_{i}}x_{a_{ij}}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $F_{*}(x_{1},\dots,x_{n})=\sum_{i=1}^{k}\prod_{j=1}^{d-b_{i}}x_{a_{ij}}$
+\end_inset
+
+ y 
+\begin_inset Formula 
+\begin{eqnarray*}
+(F_{*})^{*}(x_{1},\dots,x_{n+1}) & = & \sum_{i=1}^{k}x_{n+1}^{\max\{d-b_{i}\}}\prod_{j=1}^{d-b_{i}}\frac{x_{a_{ij}}}{x_{n+1}}=\sum_{i=1}^{k}x_{n+1}^{d-\min\{b_{i}\}-d+b_{i}}\prod_{j=1}^{d-b_{i}}x_{a_{ij}}\\
+ & = & \frac{1}{x_{n+1}^{\min\{b_{i}\}}}\sum_{i=1}^{k}x_{n+1}^{b_{i}}\prod_{j=1}^{d-b_{i}}x_{a_{ij}}=\frac{F}{x_{n+1}^{\min\{b_{i}\}}}
+\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $f\in\mathbb{K}[x,y]$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal L}:=\{(x,y)\in\mathbb{A}^{2}(\mathbb{K}):f(x,y)=0\}$
+\end_inset
+
+, llamamos 
+\series bold
+ampliación proyectiva
+\series default
+ o 
+\series bold
+completación proyectiva
+\series default
+ de 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ a 
+\begin_inset Formula $\overline{{\cal L}}:=\{<(x,y,z)>\in\mathbb{P}^{2}(\mathbb{K}):f^{*}(x,y,z)=0\}$
+\end_inset
+
+, y para 
+\begin_inset Formula $\hat{{\cal L}}\subseteq\mathbb{P}^{2}(\mathbb{K})$
+\end_inset
+
+, la 
+\series bold
+parte afín
+\series default
+ de 
+\begin_inset Formula $\hat{{\cal L}}$
+\end_inset
+
+ es 
+\begin_inset Formula $\hat{{\cal L}}^{\text{afín}}:=\{(x,y)\in\mathbb{A}^{2}(\mathbb{K}):<(x,y,1)>\in\hat{{\cal L}}\}$
+\end_inset
+
+.
+ Vemos que para 
+\begin_inset Formula $F\in\mathbb{K}[x,y,z]$
+\end_inset
+
+ homogéneo y 
+\begin_inset Formula $\hat{{\cal L}}:=\{F(x,y,z)=0\}$
+\end_inset
+
+, 
+\begin_inset Formula $\hat{{\cal L}}^{\text{afín}}=\{(x,y):F(x,y,1)=0\}=\{(x,y):F_{*}(x,y)=0\}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\overline{\hat{{\cal L}}^{\text{afín}}}=\{<(a,b,c)>:(F_{*})^{*}(a,b,c)=0\}=\hat{{\cal L}}\cup\{<(x,y,0)>:F(x,y,0)=0\}$
+\end_inset
+
+, y si 
+\begin_inset Formula $F$
+\end_inset
+
+ no es divisible por 
+\begin_inset Formula $z$
+\end_inset
+
+ es 
+\begin_inset Formula $\overline{\hat{{\cal L}}^{\text{afín}}}=\hat{{\cal L}}$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document