From 9df783bfae94f4430c8e47330d571a8c558f5092 Mon Sep 17 00:00:00 2001 From: Enrique Artal Date: Thu, 28 Nov 2024 23:41:11 +0100 Subject: [PATCH] intro --- Mult3a.tex | 67 +++++++++++++++++++++++++++++++++++++++++++----------- 1 file changed, 54 insertions(+), 13 deletions(-) diff --git a/Mult3a.tex b/Mult3a.tex index e178afc..28f4e67 100644 --- a/Mult3a.tex +++ b/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{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. +% \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 $z$. - \medskip +\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} -A couple of remarks are in order after these four steps. +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~$0,1,\infty$. -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. +\end{itemize} -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 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 $D_z$. -The following reductions are possible to perform the Fourth Step. - - \medskip +%The following reductions are possible to perform the Fourth Step. \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..