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añadido ejemplo deformaciones

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@ -632,11 +632,11 @@ 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
@ -1028,7 +1028,7 @@ or equivalently $J_{\check \tau_0}-\frac{(3, m_3)J_{\check \sigma}s}{s+1}+J_{\
\item The dual cone $\check \tau_0$ to the face $\tau_0$ is the cone generated by the rays $$\frac{1}{(m_3,3)} (3, m_3), e_1.$$ This cone has multiplicity $\frac{m_3}{(m_3,3)}$. Its contribution to the the local topological zeta function is \item The dual cone $\check \tau_0$ to the face $\tau_0$ is the cone generated by the rays $$\frac{1}{(m_3,3)} (3, m_3), e_1.$$ This cone has multiplicity $\frac{m_3}{(m_3,3)}$. Its contribution to the the local topological zeta function is
$$J_{\check \tau_0}= \frac{m_3}{(3m_3 s + 3\nu_1 + m_3)}.$$ $$J_{\check \tau_0}= \frac{m_3}{(3m_3 s + 3\nu_1 + m_3)}.$$
\end{enumerate} \end{enumerate}
Simmilarly we have that Similarly we have
$$\zeta_f (t) = \frac{(1-t^3)(1-t^{m_3})}{(1-t^{3m_3})^{(m_3,3)}}$$ $$\zeta_f (t) = \frac{(1-t^3)(1-t^{m_3})}{(1-t^{3m_3})^{(m_3,3)}}$$
@ -1073,8 +1073,11 @@ The multiplicity of this cone is $\frac{m_2+2m_3}{(2,m_2)}$. Its contribution t
The multiplicity of this cone is $\frac{2}{(2,m_2)}$. Its contribution to the the local topological zeta function is The multiplicity of this cone is $\frac{2}{(2,m_2)}$. Its contribution to the the local topological zeta function is
$$J_{\check \tau_0}=\frac{2}{\nu_1(3m_2s + 2\nu_1+m_2)}$$ $$J_{\check \tau_0}=\frac{2}{\nu_1(3m_2s + 2\nu_1+m_2)}$$
\end{enumerate} \end{enumerate}
Simmilarly we have that Similarly we have
$$\zeta_f (t) = \frac{(1-t^3)}{(1-t^{3m_2})^{(m_2,2)}}$$ $$\zeta_f (t) = \frac{(1-t^3)}{(1-t^{3m_2})^{(m_2,2)}}$$
We remark that this case it is equivalent to $f=z(z^2 - x^{m_2})$.
\subsubsection{Third case $f= z^3 - x^{m_1}z^2 - x^{m_3}$ and $\omega = x^{\nu_1-1}dxdz$ such that $m_3 > m_1$} \subsubsection{Third case $f= z^3 - x^{m_1}z^2 - x^{m_3}$ and $\omega = x^{\nu_1-1}dxdz$ such that $m_3 > m_1$}
@ -1117,6 +1120,12 @@ The multiplicity of this cone is $\frac{2}{(2,m_2)}$. Its contribution to the
$$J_{\check \tau_0}=\frac{m_1}{\nu_1(3m_1s + \nu_1+m_1)}$$ $$J_{\check \tau_0}=\frac{m_1}{\nu_1(3m_1s + \nu_1+m_1)}$$
\end{enumerate} \end{enumerate}
Similarly we have
$$\zeta_f (t) = $$
We remark that this case is equivalent to $f=(z-x^{m_1})(z^2 - x^{m_3-m_2})$
\subsection{Trinomial forms for strata $E^\circ_{i,j}$} \subsection{Trinomial forms for strata $E^\circ_{i,j}$}
@ -1199,7 +1208,236 @@ $$\frac{1}{\nu_1\nu_2(3s+1)} - \frac{2}{(3m_2s + 2\nu_1+m_2)(3n_2s + 2\nu_2+m_2)
\subsection{} Third case $f= z^3 - x^{m_1}y^{n_1}z^2 - x^{m_3}y^{n_3}$ and such that $3m_1 \leq m_3, 3 n_1 \leq n_3$ and $3(m_1, n_1) \ne 2(m_3, n_3)$. \subsection{} Third case $f= z^3 - x^{m_1}y^{n_1}z^2 - x^{m_3}y^{n_3}$ and such that $3m_1 \leq m_3, 3 n_1 \leq n_3$ and $3(m_1, n_1) \ne 2(m_3, n_3)$.
\section{On generic projections}
% In general, the $z$-axis is not a generic direction of projection for the embedded topological class of the surface given by the Weierstrass polynomial $F=z^3-2f(x,y)z+3g(x,y)$. Therefore the discriminant of $F$ with respect to $z$ given by $f^2+g^3$ is not generic inside the same equivalence class. Let us describe a situation where it is possible to describe a generic topological deformation $F_t=z^3-2f_t(x,y)z+3g_t(x,y)$ of $F$ whose discriminant $f_t^3+g_t^2$ is generic too.
%\begin{proposition}
% Assume that the functions $f$ and $g$ admit factorizations of the form $f= x^{2a} f_1$ and $g=x^{3a}g_1$. Moreover, assume that the the multiplicity of the polynomial $F_1(x, y, z)= F(x, y, x^az_1)/x^{3a}$ at the origin is equal or less than $2$. Then, any pair of deformations $f_t$ and $g_t$ of the functions $f$ and $g$ gives a topological deformation $F_t$ of $F$ and a topological deformation $f_t^3+g_t^2$ of $f_t^3+g_t^2$. Moreover, if $f_t^3+g_t^2$ is generic, then $F_t$ is generic.
%\end{proposition}
\begin{example} Consider the Weierstrass polynomial $F=z^3-3f(x,y)z+2g(x,y)$ with $f=y^2 - x^7 + x^5y$ and $g= f^2 + (y + x^2)^6$. It defines a germ of isolated surface singularity with Milnor number $\mu=41$. The discriminant of $F$ is given by
$$f^3-g^2 = f^3 - f^4 - 2f^2(y+x^2)^6 +(y+x^2)^{12}$$
and it has the topological type given by three $(2,7)$-cusp with pairwise intersection multiplicity $16$. Its dual resolution graph is pictured (left) bellow.
\
\begin{tikzpicture}
\draw[gray, thick] (6,-2) -- (10,-2);
\draw[gray, thick] (9,-2) -- (9,0);
\filldraw[black] (9,-1) circle (2pt) node[anchor=west]{$E_6$};
\filldraw[black] (9,0) circle (2pt) node[anchor=west]{$E_7$};
\filldraw[black] (6,-2) circle (2pt) node[anchor=north]{$E_1$};
\filldraw[black] (7,-2) circle (2pt) node[anchor=north]{$E_2$};
\filldraw[black] (8,-2) circle (2pt) node[anchor=north]{$E_3$};
\filldraw[black] (9,-2) circle (2pt) node[anchor=north]{$E_5$};
\filldraw[black] (10,-2) circle (2pt) node[anchor=north]{$E_4$};
\node (Z) at (9, 0) {};
\node (A) at (9, 1) {$d$};
\node (B) at (8, 1) {$d$};
\node (C) at (10, 1) {$d$};
\draw[->] (Z) edge (A);
\draw[->] (Z) edge (B);
\draw[->] (Z) edge (C);
\draw[gray, thick] (16,-2) -- (20,-2);
\draw[gray, thick] (19,-2) -- (19,1);
\filldraw[black] (19,-1) circle (2pt) node[anchor=west]{$E_6$};
\filldraw[black] (19,0) circle (2pt) node[anchor=west]{$E_8$};
\filldraw[black] (19,1) circle (2pt) node[anchor=west]{$E_7$};
\filldraw[black] (16,-2) circle (2pt) node[anchor=north]{$E_1$};
\filldraw[black] (17,-2) circle (2pt) node[anchor=north]{$E_2$};
\filldraw[black] (18,-2) circle (2pt) node[anchor=north]{$E_3$};
\filldraw[black] (19,-2) circle (2pt) node[anchor=north]{$E_5$};
\filldraw[black] (20,-2) circle (2pt) node[anchor=north]{$E_4$};
\node (W) at (19, -2) {};
\node (D) at (18, -1) {$d_t$};
\draw[->] (W) edge (D);
\node (Y) at (19, 0) {};
\node (E) at (18, 0) {$d_t$};
\draw[->] (Y) edge (E);
\end{tikzpicture}
\
The polinomial $F_t=z^3 -3f_t(x,y)z - 2 g_t(x,y)$ with
$$f_t= f + t^2y^2 \qquad \hbox{ and } \qquad g_t = g + \frac{3}{2}tyf + t^3y^3$$ defines a $\mu$-constant topological deformation of $F$ whose discriminant $f_t^3-g_t^2$ has the following embedded topological type. The discriminant of $F_t$ has two branches. One is a $(2,7)$-cusp and the other branch has two Puiseux pairs, namely, $(2,7)$ and $(2,3)$. Their intersection multiplicity is $28$. The dual resolution graph of the discriminant is pictured (right) above.
The topological zeta function and the monodromy zeta function of $F$ and $F_t$ are given by
$$ \frac{-41}{168} \frac{s^2 - 283/123s - 304/123}{(s + 16/21) (s + 19/24) (s + 1) } \quad \hbox{ and } \quad
\Phi_1 \Phi_2 \Phi_3 \Phi_4 \Phi_6 \Phi_7 \Phi_8 \Phi_{12} \Phi_{21} \Phi_{24}$$
In particular, they both have Milnor number equal to 41. This can be computed from embedded resolutions of the meromorphic functions $\mathcal{M}$ and $\mathcal{M}_t$ with the following numerical data
$$\begin{array}{c|ccccccccc}
& E_1& E_2& E_3& E_4& E_5& E_6& E_7\\
\hline
f& 2& 4& 6& 7& 14& 15& 16\\
g& 4& 8& 12& 12& 24& 24& 24\\
d& 6&12& 18& 21& 42& 45& 48
\end{array}
\qquad \begin{array}{c|ccccccccc}
& E_1& E_2& E_3& E_4& E_5& E_6& E_7& E_8\\
\hline
f_t& 2& 4& 6& 7& 14& 14& 14& 28\\
g_t& 3& 6& 9& 11& 21& 21& 21& 42\\
d_t& 6&12& 18& 21& 42& 44& 45& 90
\end{array}$$
\
\noindent and dual resolution graphs
\
\begin{tikzpicture}
\draw[gray, thick] (6,-2) -- (10,-2);
\draw[gray, thick] (9,-2) -- (9,0);
\filldraw[black] (9,-1) circle (2pt) node[anchor=west]{$E_6$};
\filldraw[black] (9,0) circle (2pt) node[anchor=west]{$E_7$};
\filldraw[black] (6,-2) circle (2pt) node[anchor=north]{$E_1$};
\filldraw[black] (7,-2) circle (2pt) node[anchor=north]{$E_2$};
\filldraw[black] (8,-2) circle (2pt) node[anchor=north]{$E_3$};
\filldraw[black] (9,-2) circle (2pt) node[anchor=north]{$E_5$};
\filldraw[black] (10,-2) circle (2pt) node[anchor=north]{$E_4$};
\node (Z) at (9, 0) {};
\node (A) at (9, 1) {$d$};
\node (B) at (8, 1) {$d$};
\node (C) at (10, 1) {$d$};
\node (F) at (8, 0) {$f$};
\draw[->] (Z) edge (A);
\draw[->] (Z) edge (B);
\draw[->] (Z) edge (C);
\draw[->] (Z) edge (F);
\node (V) at (8, -2) {};
\node (G1) at (7.25, -1) {$g$};
\node (G2) at (7.5, -1) {$g$};
\node (G3) at (8.25, -1) {$g$};
\node (G4) at (8.5, -1) {$g$};
\draw[->] (V) edge (G1);
\draw[->] (V) edge (G2);
\draw[->] (V) edge (G3);
\draw[->] (V) edge (G4);
\draw[gray, thick] (16,-2) -- (20,-2);
\draw[gray, thick] (19,-2) -- (19,1);
\filldraw[black] (19,-1) circle (2pt) node[anchor=west]{$E_6$};
\filldraw[black] (19,0) circle (2pt) node[anchor=west]{$E_8$};
\filldraw[black] (19,1) circle (2pt) node[anchor=west]{$E_7$};
\filldraw[black] (16,-2) circle (2pt) node[anchor=north]{$E_1$};
\filldraw[black] (17,-2) circle (2pt) node[anchor=north]{$E_2$};
\filldraw[black] (18,-2) circle (2pt) node[anchor=north]{$E_3$};
\filldraw[black] (19,-2) circle (2pt) node[anchor=north]{$E_5$};
\filldraw[black] (20,-2) circle (2pt) node[anchor=north]{$E_4$};
\node (W) at (19, -2) {};
\node (D) at (18, -1) {$d_t$};
\draw[->] (W) edge (D);
\node (Y) at (19, 0) {};
\node (E) at (18, 0) {$d_t$};
\draw[->] (Y) edge (E);
\node (W) at (19, -2) {};
\node (F) at (17, -1) {$f_t$};
\node (G) at (20, -1) {$g_t$};
\draw[->] (W) edge (F);
\draw[->] (W) edge (G);
\node (V) at (20, -2) {};
\node (GG) at (21, -2) {$g_t$};
\draw[->] (V) edge (GG);
\end{tikzpicture}
\
In order to that there check that both $F$ and $F_t$ define the same embedded topological type, notice that
$$F_t(x,y,z+yt)=F(x,y,t) + 3tyz^2$$
Moreover, in the left-hand side change $z \mapsto z+ty$ is an isomorphism that does not change the discriminant with respect to $z$, nor the Milnor number. Finally, in the right-hand side we have a $\mu$-constant deformation of isolated hypersurface singularities of the form $f(x) + tg(x)$, that preserves the embedded topological type according to \cite{par:99}.
Finally, it is worth mentioning that the topological type of the discriminant of $F_t$ can be realizable as $f_1^3-g_1^2$ for another pair of functions $(f_1,g_1)$. The corresponding function $F_1= z^3-3f_1z + 2g_1$ does not belong to the embedded topological type equivalence class of $F$ and $F_t$ because their Milnor number and monodromy zeta function do not coincide. The topological and monodromy zeta function of $F_1$ are
$$ \frac{-5}{21} \frac{s - 16/5}{(s + 16/21)(s + 1)} \qquad \hbox{ and, } \qquad \Phi_1 \Phi_3 \Phi_7^2 \Phi_{21}^2 $$
In particular, the Milnor number of $F_1$ is $38$.
This can be computed from embedded resolution of the meromorphic functions $\mathcal{M}_1$ with the following numerical data
$$\begin{array}{c|ccccccccc}
& E_1& E_2& E_3& E_4& E_5& E_6& E_7& E_8\\
\hline
f_t& 2& 4& 6& 7& 14& 15& 15& 30\\
g_t& 3& 6& 9& 11& 21& 22& 23& 45\\
d_t& 6&12& 18& 21& 42& 44& 45& 90
\end{array}$$
and dual resolution graph.
\
\begin{tikzpicture}
\draw[gray, thick] (16,-2) -- (20,-2);
\draw[gray, thick] (19,-2) -- (19,1);
\filldraw[black] (19,-1) circle (2pt) node[anchor=west]{$E_6$};
\filldraw[black] (19,0) circle (2pt) node[anchor=west]{$E_8$};
\filldraw[black] (19,1) circle (2pt) node[anchor=west]{$E_7$};
\filldraw[black] (16,-2) circle (2pt) node[anchor=north]{$E_1$};
\filldraw[black] (17,-2) circle (2pt) node[anchor=north]{$E_2$};
\filldraw[black] (18,-2) circle (2pt) node[anchor=north]{$E_3$};
\filldraw[black] (19,-2) circle (2pt) node[anchor=north]{$E_5$};
\filldraw[black] (20,-2) circle (2pt) node[anchor=north]{$E_4$};
\node (W) at (19, -2) {};
\node (D) at (18, -1) {$d_t$};
\draw[->] (W) edge (D);
\node (Y) at (19, 0) {};
\node (E) at (18, 0) {$d_t$};
\draw[->] (Y) edge (E);
\node (T) at (19, -1) {};
\node (U) at (19, 1) {};
\node (FT) at (18, -0.5) {$f_1$};
\node (GT1) at (18, 1) {$g_1$};
\draw[->] (T) edge (FT);
\draw[->] (U) edge (GT1);
\node (S) at (20, -2) {};
\node (GT2) at (21, -2) {$g_1$};
\draw[->] (S) edge (GT2);
\end{tikzpicture}
\
\end{example}
\bibliographystyle{amsplain} \bibliographystyle{amsplain}
\bibliography{biblio} \bibliography{biblio}