generaldegreeHypersurfaces.tex

@@ -227,7 +227,7 @@ 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 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 $(+,\dots,+)$.
+We denote each section 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:
@@ -605,7 +605,7 @@ We have then defined $\Delta^d$, the simplicial complex with which we can identi
 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 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 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 three 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 
 \[
@@ -848,7 +848,7 @@ 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 = it commit Dc.set_base_point(Dc.all_n_simplices(0)[0])
+dc_punto = Dc.set_base_point(Dc.all_n_simplices(0)[0])
 dc_punto.fundamental_group()
  -> Finitely presented group < e0 | e0^6 >
 dc_punto.cohomology()