hojas jupyter

Mult3.tex

@@ -9,7 +9,7 @@
 \usepackage{amsthm}
 \usepackage[all]{xy}
 \usepackage{mathrsfs}
-\usepackage{enumitem} 
+\usepackage{enumitem}
 \usepackage[T1]{fontenc}
 \usepackage{color}
 \usepackage{stmaryrd}
@@ -56,7 +56,7 @@
 \renewcommand{\thepropositionx}{\Alph{propositionx}} % "letter-numbered" theorems for Introduction
 
 
-\theoremstyle{definition} 
+\theoremstyle{definition}
 \newtheorem{definition}[theorem]{Definition}
 \newtheorem{question}[theorem]{Question}
 \newtheorem{example}[theorem]{Example}
@@ -229,7 +229,7 @@
 
 %%%%%%%%%%%%%%%%%%%%%
 
-% Daniel's commands 
+% Daniel's commands
 \newcommand{\Pnr}{\mathcal{P}^n_R}
 \newcommand{\Pn}{\mathcal{P}^n_X}
 \newcommand{\OXn}{\Omega^{(n)}_{X/\KK}}
@@ -304,18 +304,23 @@
 \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 $F$. 
+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. 
+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 
+
+
+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 $M_1 \nmid M_2$. 
+
+\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.
 
 \
@@ -346,6 +351,7 @@ In this case, the Newton polyhedra of $F$ contains a unique two dimensional comp
 \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);
@@ -384,14 +390,16 @@ In this case, the Newton polyhedra of $F$ contains a unique two dimensional comp
 \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. 
+\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.
 
  \
 
@@ -440,9 +448,11 @@ In this case, the Newton polyhedra of $F$ contains a unique two dimensional comp
 
 
 
+
 \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);
@@ -512,6 +522,7 @@ In this case, the Newton polyhedra of $F$ contains a unique two dimensional comp
 \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);
@@ -562,7 +573,7 @@ In this case, the Newton polyhedra of $F$ contains a unique two dimensional comp
 
 
 
-\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 
+\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 $3c_1 \ne 2c_2$.
 
@@ -570,9 +581,11 @@ Therefore, $F$ is non degenerate with respect to its Newton polyhedron if and on
 \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);
@@ -592,7 +605,9 @@ Therefore, $F$ is non degenerate with respect to its Newton polyhedron if and on
 \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);
@@ -628,8 +643,8 @@ Therefore, $F$ is non degenerate with respect to its Newton polyhedron if and on
 
 
 \end{itemize}
- 
+
- 
+
 
 
 
@@ -638,4 +653,4 @@ Therefore, $F$ is non degenerate with respect to its Newton polyhedron if and on
 
 
 
-\end{document}
+\end{document}