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Alba Larraya Sancho 2026-09-14 11:06:57 +02:00
parent 949be6632e
commit 4dfa68a341
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8
generaldegreeHypersurfaces.tex
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@ -83,10 +83,10 @@ Therefore, we will work with $C_d$, from now on.
We will denote: We will denote:
\begin{eqnarray*} \begin{eqnarray*}
M_d&=&\{(x_0,\dots,x_n)\in\CC^{n+1} | \sum x^d_i=1\} \\ M_d&=&\left\{(x_0,\dots,x_n)\in\CC^{n+1}\, \middle|\, \sum x^d_i=1\right\} \\
S_d&=&\{(x_0,\dots,x_n)\in M_d | x^d_i\in\RR_{\geq0} \forall i\} \\ S_d&=&\left\{(x_0,\dots,x_n)\in M_d\,\middle|\, x^d_i\in\RR_{\geq0} \forall i\right\} \\
K &=&\pi(S_d) K &=&\pi(S_d)
%)\{[x_0:\dots:x_{n}]\in\CC\PP^n \setminus M_2 | x^d_i\in\RR_{\geq0} \forall i\} %\{[x_0:\dots:x_{n}]\in\CC\PP^n \setminus M_2 | x^d_i\in\RR_{\geq0} \forall i\}
\end{eqnarray*} \end{eqnarray*}
where $\proy{}:\CC^{n+1}\setminus\{\mathbf{0}\}\to\PP^n$ is the quotient map. where $\proy{}:\CC^{n+1}\setminus\{\mathbf{0}\}\to\PP^n$ is the quotient map.
We then have a commutative diagram: We then have a commutative diagram:
@ -463,7 +463,7 @@ We denote as $\Delta$ the $\Delta-$complex $[v_0, \dots, v_n]$ that has $n+1$ ve
\begin{tikzcd} \begin{tikzcd}
\Delta \rar[leftrightarrow]& S_1&\\ \Delta \rar[leftrightarrow]& S_1&\\
\Delta^d\uar\rar[leftrightarrow]&S_d\uar\rar[hookrightarrow]\dar& M_d\dar\\ \Delta^d\uar\rar[leftrightarrow]&S_d\uar\rar[hookrightarrow]\dar& M_d\dar\\
&K \rar[hookrightarrow]& \PP^2\setminus C_d, &K \rar[hookrightarrow]& \PP^n\setminus C_d,
\end{tikzcd} \end{tikzcd}
\end{equation*} \end{equation*}
where $\Delta^d$ is a $\Delta-$complex that we are trying to find and identify with $S_d$, equivalently as what we have done for $d=1$. Since $S_d$ is a ramified cover of $S_1$ by the power to the $d$ map, we want $\Delta^d$ to be a ramified cover of $\Delta$ by the same map. where $\Delta^d$ is a $\Delta-$complex that we are trying to find and identify with $S_d$, equivalently as what we have done for $d=1$. Since $S_d$ is a ramified cover of $S_1$ by the power to the $d$ map, we want $\Delta^d$ to be a ramified cover of $\Delta$ by the same map.