From b844bb5faa7e2a1b099f0df7b04e3c0fb59e3bbb Mon Sep 17 00:00:00 2001 From: "m.gonzalez" Date: Thu, 8 Feb 2024 17:13:31 +0100 Subject: [PATCH] actualizar Mult3.tex --- Mult3.tex | 641 ++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 641 insertions(+) create mode 100644 Mult3.tex diff --git a/Mult3.tex b/Mult3.tex new file mode 100644 index 0000000..d6ca610 --- /dev/null +++ b/Mult3.tex @@ -0,0 +1,641 @@ +\documentclass[12pt]{amsart} +\usepackage[colorlinks=true,pagebackref,hyperindex,citecolor=Green,linkcolor=Blue]{hyperref} +\usepackage[top=1in, bottom=1in, left=1.1in, right=1.1in]{geometry} +%\usepackage{stmaryrd} +%\usepackage[firstpage]{draftwatermark} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{amsthm} +\usepackage[all]{xy} +\usepackage{mathrsfs} +\usepackage{enumitem} +\usepackage[T1]{fontenc} +\usepackage{color} +\usepackage{stmaryrd} +\usepackage{bigints} + +\usepackage{graphicx} %This is for inserting figures and 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Section \ref{counterex}} +\newtheorem*{notationInj}{Notation for Section \ref{Ln&id}} + +\newtheorem{theoremx}{Theorem} +\renewcommand{\thetheoremx}{\Alph{theoremx}} % "letter-numbered" theorems for Introduction + +\newtheorem{propositionx}{Proposition} +\renewcommand{\thepropositionx}{\Alph{propositionx}} % "letter-numbered" theorems for Introduction + + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{question}[theorem]{Question} +\newtheorem{example}[theorem]{Example} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{subsection} + +\newtheorem{questionx}[theoremx]{Question} +\renewcommand{\thequestionx}{\Alph{questionx}} % "letter-numbered" questions for Introduction + + +% MATH ------------------------------------------------------------------- + +%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%% + +% EDITING ------------------------------------------------------------------- + +% For author editorial notes 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+\newcommand{\Nn}{\overline{\Nash_n(X)}} +\newcommand{\I}{{\mathcal I}_X} +\newcommand{\V}{\mathbf{V}} + +\newcommand{\Grade}{\operatorname{grade}} +\newcommand{\Pri}{{\mathcal P}^n_X} +\newcommand{\Prix}{{\mathcal P}^n_{\Ox_{X,x}}} +\newcommand{\Prir}{{\mathcal P}^n_{R}} +\newcommand{\Pric}{{\mathcal P}^n_{\Oc}} + + +\newcommand{\T}{\mathbb T} +\newcommand{\A}{{\mathcal A}^Tw} + + +\newcommand{\GrFan}{\operatorname{GF}} +\newcommand{\XGF}{X_{\GrFan(J_n)}} +\newcommand{\GF}{\GrFan(J_n)} +\newcommand{\Nw}{\mathcal{N}_{\sigma}(\mathcal{I})} +\newcommand{\sd}{\check{\sigma}} +\newcommand{\Si}{\mbox{Sing}} +\newcommand{\Co}{\mbox{Conv}} + +\newcommand{\sz}{\sigma_{\ZZ}} +\newcommand{\Suc}{\CC[u,u^3v^4,uv]} +\newcommand{\Suk}{\KK[u,u^3v^4,uv]} +\newcommand{\Suz}{\ZZ[u,u^3v^4,uv]} +\newcommand{\Jn}{\langle u-1,u^3v^4-1,uv-1 \rangle^{n+1}} +\newcommand{\Gn}{\mathbb{G}_n} +\newcommand{\Mn}{\mathbb{M}_n} +\newcommand{\cV}{\mathcal{V}} +\newcommand{\Gc}{\mathbb{G}_n^0} +\newcommand{\Gp}{\mathbb{G}_n^p} +\newcommand{\Pnn}{\mathcal{P}_n} +\newcommand{\Pnu}{\mathcal{P}_{n-1}} +\newcommand{\Cc}{C_{\mathbb{G}_n^{(0)}}} +\newcommand{\Cp}{C_{\mathbb{G}_n^{(p)}}} +\newcommand{\bfg}{{\bf \left(\frac{f}{g}\right)^s}} +\newcommand{\bfga}{{\bf \frac{f^s}{g^{s+\alpha}}}} +\newcommand{\bfgt}{{\bf \frac{\tilde{f}^s}{\tilde{g}^{s+\alpha}}}} +\newcommand{\bfgOne}{{\bf \frac{f^s}{g^{s+1}}}} + +\newcommand{\ga}{{\bf \frac{1}{g^{\alpha}}}} +\newcommand{\gd}{\operatorname{gl.dim}} + +\newcommand{\lm}{\operatorname{lm}} +\newcommand{\lc}{\operatorname{lc}} +\newcommand{\lt}{\operatorname{lt}} +\newcommand{\init}{\operatorname{in}} +\newcommand{\diff}{\operatorname{diff}} + + +%------------------------------------------------------- + +\begin{document} + +\title{On germs of hypersurfaces with multiplicity 3} + +\author{EAB, MGV, ELC} + +\maketitle + + +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 polynomial $F = z^3 + a_1z + a_2z + a_3 \in \CC[[{\bf x}]][z]$ of degree 3 with respect to $z$. By means of a Tschirnhaus transformation, reduce $F$ to a monic equation $z^3 + b_2z + b_3$ that does not have terms of degree 2 with respect to $z$. Secondly, consider an embedded resolution $\pi: X \rightarrow \CC^d$ of the reduced divisor given by the product $b_2b_3$ of the coefficients of degree one and zero with respect to $z$. Consider the usual stratification of the pair $(X, (\pi \circ F)^{-1} \{F=0\})$ defined in terms of decomposition $\cup_{j \in J} E_j$ of $(\pi \circ F)^{-1} \{F=0\}$ into irreducible components. Pulling back $F$ under the map $\pi \times id$, we obtain for each stratum $E_I^\circ$ a quasi-trinomial local expression for $F$. +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. + +\section{The quasi-trinomial case} +In this section we consider the case of +$$F = z^3 + M_1(c_1+A)z + M_2(c_2 + M_2) \in \CC[[{\bf x}]][z]$$ +where $M_1={\bf x}^{m_1}$ and $M_2={\bf x}^{m_2}$ are monomials, $c_1,c_2 \in \CC^\ast$ and $A, B \in ({\bf x})\CC[[{\bf x}]]$ + +We distinguish the following two cases +\begin{itemize} + +\item Assume that the monomials $M_1$ and $M_2$ are not related by the divisibility relation, i.e., neither $M_1 \nmid M_2$ nor $M_1 \nmid M_2$. +In this case, the Newton polyhedra of $F$ contains a unique two dimensional compact face determined by the monomials $z^3$, $M_1z$ and $M_2$. Moreover, $F$ is non degenerate with respect to its Newton polyhedra. + +\ + + +\begin{tikzpicture} +\fill [yellow] (1,0) rectangle (3,-2); +\fill [yellow] (-2,0) rectangle (1,2); +\draw[gray, thick] (-2,-2) -- (-2,2); +\draw[gray, thick] (-2,-2) -- (3,-2); +\draw[gray, thick] (-2,0) -- (3,0); +\draw[gray, thick] (1,-2) -- (1,2); +\filldraw[black] (-2,-2) circle (2pt) node[anchor=west]{}; +\filldraw[black] (1,0) circle (2pt) node[anchor=south west]{$m_2$}; +\filldraw[black] (2,-1) circle (2pt) node[anchor=south west]{$m_1$}; + + + + + + + + + + + + +\fill [yellow] (11,0,4)--(11,0,0)--(13,0,0)--(13,0,4)--(11,0,4); +\fill [yellow] (11,0,4)--(11,0,7)--(6,0,7)--(6,0,4)--(11,0,4); + +\draw[gray, thick] (6,0,0) -- (6,0,7); +\draw[gray, thick] (6,0,0) -- (6,4,0); +\draw[gray, thick] (6,0,0) -- (13,0,0); + + + + + +\filldraw[black] (6,0,0) circle (2pt) node[anchor=west]{}; + + +\draw[gray, thick] (6,0,4) -- (13,0,4); +\draw[gray, thick] (11,0,0) -- (11,0,7); + + + +\draw[gray, thick] (6,3,0) -- (11,0,4); +\draw[gray, thick] (9,1,6) -- (11,0,4); +\draw[gray, thick] (6,3,0) -- (9,1,6); + + +\draw[gray, dashed] (9,0,0) -- (9,0,6); +\draw[gray, dashed] (6,0,6) -- (9,0,6); +\draw[gray, dashed] (9,1,0) -- (9,1,6); +\draw[gray, dashed] (6,1,6) -- (9,1,6); +\draw[gray, dashed] (9,0,6) -- (9,1,6); + +\draw[gray, dashed] (6,1,0) -- (6,1,6); +\draw[gray, dashed] (6,1,0) -- (9,1,0); +\draw[gray, dashed] (6,0,6) -- (6,1,6); +\draw[gray, dashed] (9,0,0) -- (9,1,0); + + +\filldraw[black] (6,3,0) circle (2pt) node[anchor=south west]{$z^3$}; +\filldraw[black] (11,0,4) circle (2pt) node[anchor=north west]{$M_2$}; +\filldraw[black] (9,0,6) circle (2pt) node[anchor=north]{$M_1$}; +\filldraw[black] (9,1,6) circle (2pt) node[anchor=south west]{$M_1z$}; + +} +\end{tikzpicture} + +\ + + + +\item Assume that the monomials $M_1$ and $M_2$ are related by the divisibility relation, i.e., either $M_1 | M_2$ or $M_1 | M_2$. In this case the Newton polyhedron of $F$ is a polygonal path. We distinguish the following cases. + + \ + + +\begin{tikzpicture} +\fill [yellow] (6,-2) rectangle (9,0); +\fill [yellow] (9,0) rectangle (11,2); +\draw[gray, thick] (6,-2) -- (6,2); +\draw[gray, thick] (6,-2) -- (11,-2); +\draw[gray, thick] (6,0) -- (11,0); +\draw[gray, thick] (9,-2) -- (9,2); +\filldraw[black] (6,-2) circle (2pt) node[anchor=west]{}; +\filldraw[black] (9,0) circle (2pt) node[anchor=north west]{$m_2$}; +\filldraw[black] (9.5, 1.3) circle (2pt) node[anchor=north west]{$m_1$}; +\end{tikzpicture} + +\ + + +\begin{enumerate} +\item If the point $3m_1$ does not belong to the set $2 m_2 + (\RR_{\geq 0})^2$, the Newton polyhedron of $F$ consists of two segments. The first one is determined by the monomials $z^3$ and $M_1z$ and the second is determined by the monomials $M_1z$ and $M_2$. it follows that $F$ is non degenerate with respect to its newton polyhedron. + + + + \ + + +\begin{tikzpicture} +\fill [yellow] (-2,-2) rectangle (1,-2/3); +\fill [yellow] (-2,-2) rectangle (0,0); +\draw[gray, thick] (-2,-2) -- (-2,2); +\draw[gray, thick] (-2,-2) -- (3,-2); +\draw[gray, thick] (-2,0) -- (3,0); +\draw[gray, thick] (1,-2) -- (1,2); + +\draw[gray, dashed] (0,-2/3) -- (0,0); +\draw[gray, dashed] (0,-2/3) -- (1,-2/3); + + + +\filldraw[black] (-2,-2) circle (2pt) node[anchor=west]{}; +\filldraw[black] (1,0) circle (2pt) node[anchor=north west]{$m_2$}; +\filldraw[black] (0, -2/3) circle (2pt) node[anchor=north west]{$\frac{2}{3}m_2$}; +\filldraw[black] (-0.5, -0.2) circle (2pt) node[anchor=north east]{$m_1$}; + + + + +\fill [yellow] (28/3,0,4)--(28/3,0,8/3)--(11,0,8/3)--(11,0,0)--(6,0,0)--(6,0,4)--(28/3,0,4); + + +\draw[gray, thick] (6,0,0) -- (6,0,7); +\draw[gray, thick] (6,0,0) -- (6,4,0); +\draw[gray, thick] (6,0,0) -- (13,0,0); + + + + + +\filldraw[black] (6,0,0) circle (2pt) node[anchor=west]{}; + + +\draw[gray, thick] (6,0,4) -- (13,0,4); +\draw[gray, thick] (11,0,0) -- (11,0,7); + + + +\draw[gray, thick] (8,1,3) -- (11,0,4); +\draw[gray, thick] (6,3,0) -- (8,1,3); + + +\draw[gray, dashed] (8,1,3) -- (8,0,3); +\draw[gray, dashed] (8,0,0) -- (8,0,3); +\draw[gray, dashed] (8,1,3) -- (8,1,0); +\draw[gray, dashed] (8,1,0) -- (8,0,0); +\draw[gray, dashed] (6,1,0) -- (8,1,0); + + +\draw[gray, dashed] (6,1,0) -- (6,1,3); +\draw[gray, dashed] (6,1,3) -- (8,1,3); +\draw[gray, dashed] (6,1,3) -- (6,0,3); +\draw[gray, dashed] (6,0,3) -- (8,0,3); + + + + +\draw[gray, dashed] (28/3,0,8/3) -- (28/3,0,4); +\draw[gray, dashed] (28/3,0,8/3) -- (11,0,8/3); + +\filldraw[black] (6,3,0) circle (2pt) node[anchor=south west]{$z^3$}; +\filldraw[black] (11,0,4) circle (2pt) node[anchor=north west]{$M_2$}; +\filldraw[black] (8,0,3) circle (2pt) node[anchor=north]{$M_1$}; +\filldraw[black] (8,1,3) circle (2pt) node[anchor=south west]{$M_1z$}; + + + + +\end{tikzpicture} + +\ + + + +\item If the point $3m_1$ belongs to the set $2 m_2 + (\RR_{\geq 0})^2$, the Newton polyhedron of $F$ consists of one segment with vertex corresponding to the monomials $z^3$ and $M_2$. We distinguish to further cases. + + + + + +\begin{enumerate} +\item If $M_1^3 \ne M_2^2$, the point corresponding to the monomial $M_1z$ is an interior point of the Newton polyhedron of $F$. It follows that $F$ is non degenerate with respect to its Newton polyhedron. + + + +\begin{tikzpicture} + + +\fill [yellow] (28/3,0,4)--(28/3,0,8/3)--(11,0,8/3)--(11,0,0)--(6,0,0)--(6,0,4)--(28/3,0,4); + + +\draw[gray, thick] (6,0,0) -- (6,0,7); +\draw[gray, thick] (6,0,0) -- (6,4,0); +\draw[gray, thick] (6,0,0) -- (13,0,0); + +\filldraw[black] (6,0,0) circle (2pt) node[anchor=west]{}; + + +\draw[gray, thick] (6,0,4) -- (13,0,4); +\draw[gray, thick] (11,0,0) -- (11,0,7); + + +\draw[gray, thick] (6,3,0) -- (11,0,4); + +\draw[gray, thick] (6,3,0) -- (10.5,1,3) -- (12.75, 0, 4.5); + + +\draw[gray, thick] (6,3,0) -- (6,3,7); +\draw[gray, thick] (6,3,0) -- (13,3,0); + + +\draw[gray, dashed] (6,1,3) -- (6,0,3); +\draw[gray, dashed] (10.5,0,0) -- (10.5,0,3); +\draw[gray, dashed] (6,1,3) -- (6,1,0); +\draw[gray, dashed] (10.5,1,0) -- (10.5,0,0); +\draw[gray, dashed] (6,1,0) -- (10.5,1,0); + + +\draw[gray, dashed] (10.5,1,0) -- (10.5,1,3); +\draw[gray, dashed] (10.5,1,3) -- (6,1,3); +\draw[gray, dashed] (10.5,1,3) -- (10.5,0,3); +\draw[gray, dashed] (10.5,0,3) -- (6,0,3); + + + + +\draw[gray, dashed] (28/3,0,8/3) -- (28/3,0,4); +\draw[gray, dashed] (28/3,0,8/3) -- (11,0,8/3); + +\filldraw[black] (6,3,0) circle (2pt) node[anchor=south west]{$z^3$}; +\filldraw[black] (11,0,4) circle (2pt) node[anchor=north west]{$M_2$}; +\filldraw[black] (10.5,0,3) circle (2pt) node[anchor=north]{}; +\filldraw[black] (10.5,1,3) circle (2pt) node[anchor=south west]{$M_1z$}; +\filldraw[black] (12.75,0,4.5) circle (2pt) node[anchor=north]{}; + + + +\end{tikzpicture} + + + +\item If $M_1^3 = M_2^2$, the point corresponding to the monomial $M_1z$ lies in the unique dimensional compact face of the Newton polyhedron of $F$. The face equation is equal to +$$z^3 + c_1z + c_2.$$ +Therefore, $F$ is non degenerate with respect to its Newton polyhedron if and only if $3c_1 \ne 2c_2$. + + +\begin{tikzpicture} + + +\fill [yellow] (28/3,0,4)--(28/3,0,8/3)--(11,0,8/3)--(11,0,0)--(6,0,0)--(6,0,4)--(28/3,0,4); + + +\draw[gray, thick] (6,0,0) -- (6,0,7); +\draw[gray, thick] (6,0,0) -- (6,4,0); +\draw[gray, thick] (6,0,0) -- (13,0,0); + +\filldraw[black] (6,0,0) circle (2pt) node[anchor=west]{}; + + +\draw[gray, thick] (6,0,4) -- (13,0,4); +\draw[gray, thick] (11,0,0) -- (11,0,7); + + +\draw[gray, thick] (6,3,0) -- (11,0,4); + + + +\draw[gray, thick] (6,3,0) -- (6,3,7); +\draw[gray, thick] (6,3,0) -- (13,3,0); + + +\draw[gray, dashed] (6,1,8/3) -- (6,0,8/3); +\draw[gray, dashed] (28/3,0,0) -- (28/3,0,8/3); +\draw[gray, dashed] (6,1,8/3) -- (6,1,0); +\draw[gray, dashed] (28/3,1,0) -- (28/3,0,0); +\draw[gray, dashed] (6,1,0) -- (28/3,1,0); + + +\draw[gray, dashed] (28/3,1,0) -- (28/3,1,8/3); +\draw[gray, dashed] (28/3,1,8/3) -- (6,1,8/3); +\draw[gray, dashed] (28/3,1,8/3) -- (28/3,0,8/3); +\draw[gray, dashed] (28/3,0,8/3) -- (6,0,8/3); + + + + +\draw[gray, dashed] (28/3,0,8/3) -- (28/3,0,4); +\draw[gray, dashed] (28/3,0,8/3) -- (11,0,8/3); + +\filldraw[black] (6,3,0) circle (2pt) node[anchor=south west]{$z^3$}; +\filldraw[black] (11,0,4) circle (2pt) node[anchor=north west]{$M_2$}; +\filldraw[black] (28/3,0,8/3) circle (2pt) node[anchor=north]{}; +\filldraw[black] (28/3,1,8/3) circle (2pt) node[anchor=south west]{$M_1z$}; + + + + +\end{tikzpicture} + + +\end{enumerate} +\end{enumerate} + + + + +\end{itemize} + + + + + +\bibliographystyle{amsplain} +\bibliography{biblio} + + + +\end{document} \ No newline at end of file