intro

Mult3a.tex

@@ -358,26 +358,67 @@ that $b_2b_3\neq 0$. Under this condition, up to a constant, $\disc_zF$ is $D_z:
 %We propose the following general method to study germs of multiplicity 3 singular points of hypersurfaces. Firstly, consider an equation of the germ given by a monic Weierstrass polynomial $F = z^3 + a_1z^2 + a_2z + a_3 \in \CC[[{\bf x}]][z]$ of degree 3 with respect to $z$. Assume that $D_z F$ is not zero. We will be mainly interested in the case when $F$ is degenerate with respect to its Newton polyhedron. Otherwise, the partial resolution associated with the dual of the Newton polyhedron of $F$ gives plenty information of the geometry and topology of the singular point of the hypersurface defined by $F$. By means of a Tschirnhaus transformation, reduce $F$ to a monic Weierstrass equation $z^3 + b_2z + b_3$ that does not have terms of degree 2 with respect to $z$. Assume that  the coefficients of degree one and zero with respect to $z$ are both non zero, i.e., the product $b_2b_3 \ne 0$.  
 
 \begin{paso}
-Consider an embedded resolution $\pi: X \rightarrow \CC^2$ of the reduced divisor given by the product $G=b_2b_3D_zF$. Consider the usual stratification of the pair $(X, (\pi \circ G)^{-1} \{G=0\})$ defined in terms of decomposition $\cup_{j \in J} E_j$ of $(\pi \circ G)^{-1} \{G=0\}$ into irreducible components. Pulling back $F$ under the map $\pi \times id$, we obtain for each stratum $E_I^\circ$ a \emph{quasi-trinomial local expression} for $F$. Section \ref{3nomial} clasifies the 5 posible quasi-trinomial local forms having terms of degrees 3, 1, and 0 with respect to $z$. These are the quasi-trinomials forms that appear at this step because we have previously made a Tschirnhaus transformation. Notice that the local expression is quasi-trinomial due to the fact that $\pi$ resolves both $b_2$ and $b_3$. Moreover, the quasi-trinomial local form is quasi-ordinary because $\pi$ also resolves the discriminant $D_zF$. The local quasi-trinomial expression will be non degenerate with respect to its Newton polyhedron unless the monomials $M_1$, $M_2$ and their coefficients $c_1$ and $c_2$ satisfy a precise condition. {\color{red}In principle, this precise condition can be defined on a subset of a stratum of $E_I^\circ$, with $\emptyset \ne I \subset J$. If it is the case, we consider the corresponding refinement of the stratification.} Moreover, the degeneracies of the degenerate trinomial local case are either on horizontal faces or of the form $(z-af(x,y))(z-bf(x,y))$ with $a, b \in \CC^\ast$, and $f \in \CC[x,y]$.  
+Consider an embedded resolution $\pi: X \rightarrow \CC^2$ of the reduced divisor given by the product $G=b_2b_3D_zF$. Consider the usual stratification of the pair $(X, (\pi \circ G)^{-1}(0))$ 
+defined in terms of decomposition $\bigcup_{j \in J} E_j$ of $(\pi \circ G)^{-1}(0)$ into irreducible components. 
+At any point $p\in(\pi \circ G)^{-1}(0)$ we consider the pull-back $F_p$ of $F$ under
+the map $\pi \times \id$. These germs satisfy the following properties:
+\begin{enumerate}[label=(PB\arabic{enumi})]
+\item\label{pb1} Under some \emph{natural} coordinates at $p$, $b_2$ and $b_3$ are monomials, up to a unit, i.e., 
+the local Newton polytope of $F_p$ is a polygon. If we consider the polynomial $\tilde{F}_p$ supported
+by the vertices of the Newton polytope, it is a \emph{trinomial}.
+We say then that $F_p$ is a \emph{quasi-trinomial}.
+
+\item Under the coordinates of \ref{pb1}, $F_p$ is quasi-ordinary with respect to $z$, since $D_z$ is a monomial
+up to a unit.
+
+\item For a given stratum $E^\circ_I$, the local Newton polytope of $F_p$ does not depend on the particular~$p$.
+In particular, the trinomials $\tilde{F}_p$ share the monomials (in $z^3, z, z^{0}$).
+
+\item For a given stratum $E^\circ_I$, the coefficients of $\tilde{F}_p$ vary, but the degenerate
+or non-degenerate character does not. In particular, if it is non-degenerate the \emph{topological properties}
+of $F_p$ coincide with the corresponding trinomial where all the coefficients equal~$1$.
+
+\item If at one stratum we have a degenerate character, there is a change of variables $z\mapsto z + h(x, y)$,
+for some suitable $h$ such that $F_p$ becomes non-degenerate under this change, actually
+of trinomial type (with monomials in $z^3, z^2, z^0$). Moreover, the numerical data
+of the resolution for $b_2,b_3,D_z$ determines the monomials.
+\end{enumerate}
+In Section~\ref{3nomial} we will study the classes of possible quasi-trinomial local forms.
+
 \end{paso}
 
-\noindent \emph{Third step:} Make translations of the type $z \mapsto z + h(x,y)$, where $h(x,y)$ is determined by the local quasi-ordinary expansion, to obtain a new set of coordinates such that the local form is non degenerate. Notice that after the translation terms of degree 2 respect to $z$ appear.  In fact, we obtain a new type of quasi-trinomial local form with terms of degree $3$, $2$, and $0$ with respect to $z$.
+% having terms of degrees 3, 1, and 0 with respect to $z$. 
+% These are the quasi-trinomials forms that appear at this step because we have previously made a Tschirnhaus transformation. Notice that the local expression is quasi-trinomial due to the fact that $\pi$ resolves both $b_2$ and $b_3$. Moreover, the quasi-trinomial local form is quasi-ordinary because $\pi$ also resolves the discriminant $D_zF$. The local quasi-trinomial expression will be non degenerate with respect to its Newton polyhedron unless the monomials $M_1$, $M_2$ and their coefficients $c_1$ and $c_2$ satisfy a precise condition. {\color{red}In principle, this precise condition can be defined on a subset of a stratum of $E_I^\circ$, with $\emptyset \ne I \subset J$. If it is the case, we consider the corresponding refinement of the stratification.} Moreover, the degeneracies of the degenerate trinomial local case are either on horizontal faces or of the form $(z-af(x,y))(z-bf(x,y))$ with $a, b \in \CC^\ast$, and $f \in \CC[x,y]$.  
 
- \medskip
+% \noindent \emph{Third step:} Make translations of the type $z \mapsto z + h(x,y)$, where $h(x,y)$ is determined by the local quasi-ordinary expansion, to obtain a new set of coordinates such that the local form is non degenerate. Notice that after the translation terms of degree 2 respect to $z$ appear.  In fact, we obtain a new type of quasi-trinomial local form with terms of degree $3$, $2$, and $0$ with respect to $z$.
 
-\noindent \emph{Fourth step:} Use the stratification principle with the stratification introduced in the Second Step, the quasi-trinomial local forms and, if needed, the corresponding local differential forms to compute \emph{additive / multiplicative local invariants} of the singular point defined by $f$ at $o$. For instance, we will interested in the A'Campo zeta function and in the topological zeta function.
+\begin{paso}
+Using the stratification principle of~\cite{ACNLM02}, 
+the quasi-trinomial local forms and, the corresponding local differential forms we compute 
+some \emph{additive} or \emph{multiplicative} local invariants of the singular point defined by $f$ at $0$. 
+For instance, we will interested in the A'Campo zeta function and in the topological zeta function.
+\end{paso}
 
- \medskip
+A couple of remarks are in order after these steps. Note that the resolution~$\pi$ is also a resolution
+of the meromorphic function defined by $\frac{b_2^3}{b_3^2}$, since we \emph{separate} three fibers;therefore, 
+the irreducible components $E_j$  of $(\pi \circ G)^{-1} (0)$ 
+can be classified into~$5$ different types according to the image of $E_j$ under the morphism $\mathcal{M}:X\to\mathbb{P}^1$ 
+induced by  $\frac{b_2^3}{b_3^2}$. We distinguish the following types of components:
+\begin{itemize}
+\item The zeros are the components $E_j$ such that $\mathcal{M}_{|E_j}$ is the constant map to~$0$.
+\item The poles are the components $E_j$ such that $\mathcal{M}_{|E_j}$ is the constant map to~$\infty$.
+\item The dicritical components are the $E_j$ such that $\mathcal{M}_{|E_j}$ is surjective.
+\item The discriminant compoents are the $E_j$ such that $\mathcal{M}_{|E_j}$ is the constant map to~$1$.
+\item The constant components are the $E_j$ such that $\mathcal{M}_{|E_j}$ is the constant map to some
+element distinct to~$0,1,\infty$.
 
-A couple of remarks are in order after these four steps. 
+\end{itemize}
 
-First, these steps achieve a resolution of the meromorphic function defined by $\frac{b_2^3}{b_3^2}$ and therefore the irreducible components $E_j$  of $(\pi \circ F)^{-1} \{F=0\}$ can be classified into 5 different types according to the image of $e_j$ under the meromorphic function $\mathcal{M}$ defined by  $\frac{b_2^3}{b_3^2}$. There are components $E_j$ that are mapped to $0$ called zeros, components $E_j$ that are mapped to $\infty$ called poles, components $E_j$ that are mapped surjectively onto $\mathbb{P}^1$ called dicritical, components that are mapped to the value 1, and finally components that are mapped to other values different  than 0, 1, and $\infty$. The image of a component $E_j$ is determined by the orders of vanishing of  $f$, $g$ and $\mathcal{M}= \frac{b_2^3}{b_3^2}$ along the divisor $E_j$ and (in some cases by) their pattern of intersection. 
+The image of a component $E_j$ is determined by the orders of vanishing of  $f$, $g$ and $\mathcal{M}= \frac{b_2^3}{b_3^2}$ along the divisor $E_j$ and (in some cases by) their pattern of intersection. 
+With the same ideas 
+the trinomial part of any quasi-trinomial local form, given by the formal restriction of the quasi-trinomial form to its Newton polyhedron, is determined by the valuations associated with $f$, $g$ and $D_z$. 
 
-Second, the trinomial part of any quasi-trinomial local form, given by the formal restriction of the quasi-trinomial form to its Newton polyhedron, is determined by the valuations associated with $f$, $g$ and $\mathcal{M}= \frac{b_2^3}{b_3^2}$ too. 
+%The following reductions are possible to perform the Fourth Step.
-
-The following reductions are possible to perform the Fourth Step.
-
- \medskip
 
 \noindent \emph{Reduction A:} It is enough to consider the associated trinomial local forms instead of the quasi-trinomial local forms.
 
@@ -1366,7 +1407,7 @@ Z(F_7, \omega_7)_{o_7}\!=\!Z(F_{7,33}, \omega_{7,33})_{o_{7,33}}\!=\! Z(F_{15},
 \]
 \end{example}
 
-  \subsection{Second reduction: Recursive elimination of exceptional  \texorpdfstring{$(-1)$}{-1}-curve not intersecting the strict transform of the discrimiant}
+  \subsection{Second reduction: Recursive elimination of exceptional  \texorpdfstring{$(-1)$}{-1}-curve not intersecting the strict transform of the discriminant}
   
   After performing the First reduction, exceptional $(-1)$-curves that do not intersect the strict transform of the discriminant may appear. The next proposition shows that we can eliminante these divisors..