Compilar

Mult3.tex

@@ -572,7 +572,7 @@ $b_3= (y^3+x^4)(x^3+y^4)$. The minimal resolution $\pi$ of the product $G=b_2 b_
 &E_1&E_2, E_{10}&E_3, E_{11}&E_4, E_{12}&E_5, E_{13}&E_6, E_{14}&E_7, E_{15} &E_8, E_{16}&E_9, E_{17}\\
 \hline
 \hline
-N^f_i& {\bf 8}& 10& 20& 28& 36& 44& \bf 52}& 21&42\\
+N^f_i& {\bf 8}& 10& 20& 28& 36& 44& {\bf 52}& 21&42\\
 N^g_i& {\bf12}& {\bf 13}& {\bf 26}& {\bf 39}& {\bf 52}& {\bf 65}& {\bf 78}& {\bf 26}& {\bf 52}\\
 N^\Delta_i& 25& {\bf 26}& {\bf 52}& {\bf 78}& {\bf 104}& {\bf 130}& {\bf 156}& {\bf 52}&{\bf 104}\\
 \hline
@@ -620,7 +620,7 @@ Since $\pi$ is a resolution of both $b_2$ and $b_3$, over each non empty strata
 \end{equation}
 where $u_1, u_2$  are units in $\CC\{x_1, y_1\}$. Hence, we have expressions 
 \begin{equation}\label{quasi-trinomial}
-z^3 -3 x_1^{N^0_{i_1}} y_1^{N^0_{i_2}} u_1(x_1, y_1) z +  2 x_1^{N^\infty_{i_1}} y_1^{N^\infty_{i_2}}  u_2(x_1,y_1),\end{equation}
+z^3 -3 x_1^{N^0_{i_1}} y_1^{N^0_{i_2}} u_1(x_1, y_1) z +  2 x_1^{N^\infty_{i_1}} y_1^{N^\infty_{i_2}}  u_2(x_1,y_1),
 \end{equation}
 for each non empty strata $E^\circ_I$. Notice that these expressions are quasi-ordinary because $\pi$ is also a resolution of $D_zF$ and therefore $D_z(\pi^\ast F)_{|_I}$ is the product of a monomial in the variables $x_1$ and $y_1$ and a unit in $\CC\{x_1, y_1\}$. 
 
@@ -632,13 +632,13 @@ If the strata $E^\circ_{i,j} \ne \emptyset$ and $\mathcal{M}(E_i)=1$ then either
 
 
 \begin{proposition} There exist a change of coordinates of the form
-$$x_1= x_2, \quad y_1= y_2, \quad z = z_1 - x_2^{\frac{N^0_i}{2}}y_2^{\frac{N^0_j}{2}}} + \cdots$$
+$$x_1= x_2, \quad y_1= y_2, \quad z = z_1 - x_2^{\frac{N^0_i}{2}}y_2^{\frac{N^0_j}{2}} + \cdots$$
  such that (\ref{quasi-trinomial}) becomes
  either
-$$z_1^3 + x_2^{\frac{N^0_i}{2}}y_2^{\frac{N^0_j}{2}}} u'_1(x_2,y_2) z_1^2   + x_2^{N_i^\infty + M_i^\Delta}y_2^{N_i^\infty + M_i^\Delta} u'_2(x_2,y_2)$$
-if both  $\mathcal{M}(E_i) =  $\mathcal{M}(E_j)=1$
+$$z_1^3 + x_2^{\frac{N^0_i}{2}}y_2^{\frac{N^0_j}{2}} u'_1(x_2,y_2) z_1^2   + x_2^{N_i^\infty + M_i^\Delta}y_2^{N_i^\infty + M_i^\Delta} u'_2(x_2,y_2)$$
+if both  $\mathcal{M}(E_i) = \mathcal{M}(E_j)=1$
 or
-$$z_1^3 + x_2^{\frac{N^0_i}{2}}y_2^{\frac{N^0_j}{2}}} u'_1(x_2,y_2) z_1^2   + x_2^{N_i^\infty + M_i^\Delta}y_2^{N_j^\infty } u'_2(x_2,y_2)$$
+$$z_1^3 + x_2^{\frac{N^0_i}{2}}y_2^{\frac{N^0_j}{2}} u'_1(x_2,y_2) z_1^2   + x_2^{N_i^\infty + M_i^\Delta}y_2^{N_j^\infty } u'_2(x_2,y_2)$$
 if  $\mathcal{M}(E_i)=1$ and  $E_j$ is dicritical
 \end{proposition}