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\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}