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Edwin León 2024-05-14 13:08:13 +02:00
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Mult3.tex
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@ -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}\\ &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
\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^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}\\ N^\Delta_i& 25& {\bf 26}& {\bf 52}& {\bf 78}& {\bf 104}& {\bf 130}& {\bf 156}& {\bf 52}&{\bf 104}\\
\hline \hline
@ -620,7 +620,7 @@ Since $\pi$ is a resolution of both $b_2$ and $b_3$, over each non empty strata
\end{equation} \end{equation}
where $u_1, u_2$ are units in $\CC\{x_1, y_1\}$. Hence, we have expressions where $u_1, u_2$ are units in $\CC\{x_1, y_1\}$. Hence, we have expressions
\begin{equation}\label{quasi-trinomial} \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} \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 $\CC\{x_1, y_1\}$. 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 \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 such that (\ref{quasi-trinomial}) becomes
either either
$$z_1^3 + x_2^{\frac{N^0_i}{2}}y_2^{\frac{N^0_j}{2}}} u'_1(x_2,y_2) z_1^2 + x_2^{N_i^\infty + M_i^\Delta}y_2^{N_i^\infty + M_i^\Delta} u'_2(x_2,y_2)$$ $$z_1^3 + x_2^{\frac{N^0_i}{2}}y_2^{\frac{N^0_j}{2}} u'_1(x_2,y_2) z_1^2 + x_2^{N_i^\infty + M_i^\Delta}y_2^{N_i^\infty + M_i^\Delta} u'_2(x_2,y_2)$$
if both $\mathcal{M}(E_i) = $\mathcal{M}(E_j)=1$ if both $\mathcal{M}(E_i) = \mathcal{M}(E_j)=1$
or 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 if $\mathcal{M}(E_i)=1$ and $E_j$ is dicritical
\end{proposition} \end{proposition}