diff --git a/Mult3.pdf b/Mult3.pdf index c9602f3..00312a5 100644 Binary files a/Mult3.pdf and b/Mult3.pdf differ diff --git a/Mult3.tex b/Mult3.tex index d287bac..f049320 100644 --- a/Mult3.tex +++ b/Mult3.tex @@ -310,13 +310,13 @@ Consider a the germ of a multiplicity 3 singular point of hypersurface. An equat \medskip -\noindent \emph{First step:} By means of a Tschirnhaus transformation, reduce $F$ to a monic Weierstrass equation $F = z^3 -3 b_2z + 2 b_3$ 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 $b_2b_3 \ne 0$. If either $b_2 =0$ and $b_3 \ne 0$ or $b_2 \ne 0$ and $b_3=0$ it reduces to the ase of a suspension that we have studied elsewhere. See \cite{}. +\noindent \emph{First step:} By means of a Tschirnhaus transformation, reduce $F$ to a monic Weierstrass equation $F = z^3 -3 b_2z + 2 b_3$ 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 $b_2b_3 \ne 0$. If either $b_2 =0$ and $b_3 \ne 0$ or $b_2 \ne 0$ and $b_3=0$ it reduces to the case of a suspension that we have studied elsewhere. See \cite{}. %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 $b_2b_3 \ne 0$. \medskip -\noindent \emph{Second step:} 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. 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 $f \in \CC[x,y]$. +\noindent \emph{Second step:} 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 $f \in \CC[x,y]$. \medskip @@ -402,6 +402,8 @@ The monic Weierstrass polynomial $z^3 -3b_2z + 2b_3$ is non degenerate unless $M \section{Jung´s method for power series of multiplicity three } +\subsection{Embbeded resolution of the discriminant and classification of their exceptional divisors} + 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, i.e., for each subset $I \subset J$ we define $$E_I^\circ = \bigcap_{i \in I} E_j \setminus \bigcup_{j \not \in I} E_j .$$ @@ -421,12 +423,17 @@ There are four possibilities for the behaviour of the numbers $N^0_j, N^\infty_j \item $3 N^0_j = 2N^\infty_j < N^\Delta_j$ and in this case it is mapped constantly to $1$ under $\mathcal{M}$. \end{itemize} -In order to distinguish between dicritical divisors and divisors that are mapped constantly to a value different than $0$, $1$ or $\infty$ one notices that connected components of dicritical divisors must intersect both to zeros and poles and connected components of divisors mapped constantly to a value different than $0$, $1$ or $\infty$ do not intersect zeros or poles but do intersect dicritical divisors. + +\begin{remark} +In the third case, $N^\Delta_j$ is divisible by 6. In order to distinguish between dicritical divisors and divisors that are mapped constantly to a value different than $0$, $1$ or $\infty$, one notices that connected components of dicritical divisors must intersect both to zeros and poles and connected components of divisors mapped constantly to a value different than $0$, $1$ or $\infty$ do not intersect zeros or poles but do intersect dicritical divisors. + + +If the strata $E^\circ_{i,j} \ne \emptyset$ and $\mathcal{M}(E_i)=1$ then either $\mathcal{M}(E_j)=1$ or $E_j$ is dicritical, i.e., $\mathcal{M}(E_j)=\mathbb{P}^1$. +\end{remark} - -\begin{definition} For a divisor of the last type, the difference +\begin{definition}\label{extriplepole} For a divisor of th last type, i.e., a divisor that is mapped constantly to $1$, the difference $N^\Delta_j - 3 N^0_j = N^\Delta_j - 2 N^\infty_j$ is called the \emph{excess of the discriminant} and it is denoted by $M^\Delta_j$. \end{definition} @@ -435,7 +442,7 @@ $N^\Delta_j - 3 N^0_j = N^\Delta_j - 2 N^\infty_j$ is called the \emph{excess of \begin{example} Consider $F=z^3 - 3b_2z + 2b_3$ with $b_2= ((y^2 + x^3)^2 + x^5y)((x^2 + y^3)^2 + y^5x)$ and -$b_3= (y^3+x^4)(x^3+y^4)$. The minimal resolution $\pi$ of the product $G=b_2 b_3 D_zF$ is obtained after 15 blow-ups. The following figure and table give the dual resolution graph of $G$ and its associated numerical data. +$b_3= (y^3+x^4)(x^3+y^4)$. The minimal resolution $\pi$ of the product $G=b_2 b_3 D_zF$ is obtained after 30 blow-ups. The following figure and table give the dual resolution graph of $G$ and its associated numerical data. @@ -453,7 +460,7 @@ $b_3= (y^3+x^4)(x^3+y^4)$. The minimal resolution $\pi$ of the product $G=b_2 b_ \coordinate (G5) at (-0.1,0.75); \coordinate (G6) at (-0.2,0.75); \coordinate (G7) at (0,0.75); - \node[above] at (G7) {{\small ${{F^\Delta_i}}_{{}_{i=1}}^{{}{13}}$}}; + \node[above] at (G7) {{\small ${{F^\Delta_i}}_{{}_{i=16}}^{{}{28}}$}}; \coordinate (G8) at (0.1,0.75); \coordinate (G9) at (0.2,0.75); \coordinate (G10) at (0.3,0.75); @@ -485,17 +492,17 @@ $b_3= (y^3+x^4)(x^3+y^4)$. The minimal resolution $\pi$ of the product $G=b_2 b_ \coordinate (F1) at (1.2,0.75); - \node[above] at (F1) {{\small $F^{\infty}_1$}}; + \node[above] at (F1) {{\small $F^{\infty}_{33}$}}; \coordinate (F2) at (1.95,0.75); - \node[above] at (F2) {{\small $F^\Delta_{14}$}}; + \node[above] at (F2) {{\small $F^\Delta_{29}$}}; \coordinate (F3) at (6,0.75); - \node[above] at (F3) {{\small $F^{0}_1$}}; + \node[above] at (F3) {{\small $F^{0}_{31}$}}; \coordinate (F4) at (-1.2,0.75); - \node[above] at (F4) {{\small $F^{\infty}_2$}}; + \node[above] at (F4) {{\small $F^{\infty}_{34}$}}; \coordinate (F5) at (-1.95,0.75); - \node[above] at (F5) {{\small $F^\Delta_{15}$}}; + \node[above] at (F5) {{\small $F^\Delta_{30}$}}; \coordinate (F6) at (-6,0.75); - \node[above] at (F6) {{\small $F^{0}_2$}}; + \node[above] at (F6) {{\small $F^{0}_{32}$}}; @@ -578,42 +585,45 @@ N^\Delta_i& 25& {\bf 26}& {\bf 52}& {\bf 78}& {\bf 104}& {\bf 130}& {\bf 156}& { \hline \hline \nu_i&2&3&5&7&9&11&13&6&11\\ -\chi()&-13&1&-1&2&2&2&-4&1&-1\\ +\chi()&-11&1&-1&0&0&0&-2&1&-1\\ E_i^2& -7& -2& -2& -2& -2& -2& -1&-2&-1\\ \hline \end{array}$$ -The divisor $E_1$ is mapped to the value 1, the divisors $E_7$ and $E_{13}$ are dicriticals, and the rest of exceptionals divisors are poles. The 13 arrows $F^\Delta_i$ with $i=1, \dots, 13$ attached to the divisor $E_1$ correspond to irreducible components of the strict transforms of $D_zF$, the arrows attachef to the divisor $E_7$ (resp. $E_{13}$) correspond to an irreducible component strict transform of $b_3$ and another irreducible component of $D_zF$. +The divisor $E_1$ is mapped to the value 1, the divisors $E_7$ and $E_{15}$ are dicriticals, and the rest of exceptionals divisors are zeros. The 13 arrows $F^\Delta_i$ with $i=16, \dots, 29$ attached to the divisor $E_1$ correspond to irreducible components of the strict transforms of $D_zF$, the arrows attached to the divisor $E_7$ (resp. $E_{13}$) correspond to an irreducible component strict transform of $b_3$ and another irreducible component of $D_zF$. Finally the arrow attched to $E_9$ (resp. to $E_{17}$) correspond to the strict transform of $b_2$. %$$\pi^{-1} \{\mathcal{M} = 0\} = 2 (F^{0}_1 + F^{0}_2) + 4 (E_2 + E_5 + E_{10} + E_{13}) + 6 (E_4 + E_{12} ) + 8 (E_3 + E_{11}) + 11 (E_8+E_{16}) + 22 (E_9 + E_{17})) , \quad \pi^{-1} \{\mathcal{M} = 1\} = \sum_{i=1}^{15} F^\Delta_i + E_1$$ -$$\pi^{-1} \{\mathcal{M} = 0\} = 2 ( \sum_{1,2}F^{0}_i + \sum_{6,14} E_i ) + 4 \sum_{2,5,10,13}E_i + 6 \sum_{4,12 }E_i + 8 \sum_{3,11}E_i + 11\sum_{8,16} E_i + 22 \sum_{9,17} E_i ,$$ - $$\pi^{-1} \{\mathcal{M} = \infty\} = 3 F^{\infty}_1 + 3 F^{\infty}_2, \quad \pi(E_7) = \pi(E_{15}) = \mathbb{P}^1, \quad \pi^{-1} \{\mathcal{M} = 1\} = \sum_{i=1}^{15} F^\Delta_i + E_1$$ +$$\pi^{-1} \{\mathcal{M} = 0\} = 2 ( \sum_{31,32}F^{0}_i + \sum_{6,14} E_i ) + 4 \sum_{2,5,10,13}E_i + 6 \sum_{4,12 }E_i + 8 \sum_{3,11}E_i + 11\sum_{8,16} E_i + 22 \sum_{9,17} E_i ,$$ + $$\pi^{-1} \{\mathcal{M} = \infty\} = 3 F^{\infty}_{33} + 3 F^{\infty}_{34}, \quad \pi(E_7) = \pi(E_{15}) = \mathbb{P}^1, \quad \pi^{-1} \{\mathcal{M} = 1\} = \sum_{i=16}^{30} F^\Delta_i + E_1$$ -The stratification associated with the resolution $\pi$ consist of $35$ strata $E^\circ_{i,j}$ of dimension 0 and 8 strata $E_i^\circ$ of dimension 1 and $\chi(E_i^\circ) \ne 0$. -The former correspond to the 16 pairs $(i,j)$ listed belllow -$$\begin{array}{cccccccc} -(1,7)& (2,3)& (3,4)& (3,9)& (4,5)& (5,6)& (6,7)& (8, 9) \\ -(1,15)& (10,11)& (11,12)& (11,17)& (12, 13)& (13,14)& (14,15)& (16,17) -\end{array}$$ -the 13 intersections $E_1 \cap F_i^\Delta$ for $i=1, \dots, 13$ and the 6 intersections -$$ E_7 \cap F_1^\infty, \quad E_7 \cap F_{14}^\Delta, \quad E_9 \cap F_1^0, \quad E_{15} \cap F_2^\infty, \quad E_{15} \cap F_{14}^\Delta, \quad E_{17} \cap F_2^0 $$ +The stratification associated with the resolution $\pi$ consist of $35$ strata $E^\circ_{i,j}$ of dimension 0 and 11 strata $E_i^\circ$ of dimension 1 and $\chi(E_i^\circ) \ne 0$. +The former correspond to the elements $(i,j)$ of the set +$$S= \left \{ \begin{array}{ccccccccc} +(1,7)& (1,15)& (1,16)& (1,17)& (1,18)& (1,19)& (1,20)& (1,21)& (1,22)\\ +(1,23)& (1,24)& (1, 25)& (1, 26)& (1, 27)& (1, 28)& (2,3)& (3,4)& (3,9)\\ +(4,5)& (5,6)& (6,7)& (7, 29)& (7, 33)& (8, 9)& (9,31)& (10,11)& (11,12)\\ + (11,17)& (12, 13)& (13,14)& (14,15)& (15,30)& (15, 34)& 16,17)& (17,32) +\end{array} \right \} $$ +%the 13 intersections $E_1 \cap F_i^\Delta$ for $i=16, \dots, 28$ and the 6 intersections +%$$ E_7 \cap F_{34}^\infty, \quad E_7 \cap F_{30}^\Delta, \quad E_9 \cap F_{32}^0, \quad E_{15} \cap F_{35}^\infty, \quad E_{15} \cap F_{31}^\Delta, \quad E_{17} \cap F_{33}^0 $$ and the latter correspond with the indexes -$$1, \quad 2, \quad 3, \quad 7, \quad 8, \quad 10, \quad 11, \quad 15.$$ +$$1, \quad 2, \quad 3, \quad 7, \quad 8, \quad 9, \quad 10, \quad 11, \quad 15, \quad 16, \quad 19.$$ Notice that the minimal resolution $\pi'$ of the discriminant $D_zF$ is obtained after just 11 blow-ups. The blowups corresponding to divisors $E_8, E_9, E_{14}$ and $E_{15}$ are needed to resolve $b_2$. \end{example} -Let us consider the birational map $\tilde{\pi} := ( \pi, id ): \tilde{X} : =X \times \CC \rightarrow \CC^{3}$, where $id: \CC \rightarrow \CC$ is the identity map, and the pullback of $F$ under $\pi \times id$. The pullback $\pi^\ast F$ can be expressed in terms of the pullbacks of $b_2$ and $b_3$ as +\subsection{Trinomial forms associated to the strata} + +Let us consider the birational map $\tilde{\pi} := ( \pi, id ): \tilde{X} : =X \times \CC \rightarrow \CC^{3}$, where $id: \CC \rightarrow \CC$ is the identity map, and the pullback of $F$ under $\pi \times id$. The pullback $\tilde{\pi}^\ast F$ can be expressed in terms of the pullbacks of $b_2$ and $b_3$ as \begin{equation} z^3 -3 \pi^\ast b_2 z + 2 \pi^\ast b_3. \end{equation} -Over each non empty strata $E^\circ_I$ of $X$ created by the resolution $\pi$, -for the pullback of $F$ under the birational map $\pi \times id$. -Since $\pi$ is a resolution of both $b_2$ and $b_3$, over each non empty strata $E^\circ_I$ of $X$ created by the resolution $\pi$, the terms $(\pi^\ast b_3)_{|_I}$ and $(\pi^\ast b_3)_{|_I}$ can be reexpressed in terms of the multiplicities $N^f_i$ and $N^g_i$ with $i \in I$ as +over each non empty strata $E^\circ_I$ of $X$ created by the resolution $\pi$. +Since $\pi$ is a resolution of both $b_2$ and $b_3$, over each non empty strata $E^\circ_I$ of $X$ created by the resolution $\pi$, the terms $(\pi^\ast b_3)_{|_I}$ and $(\pi^\ast b_3)_{|_I}$ can be reexpressed in terms of the multiplicities $N^0_i$ and $N^\infty_i$ with $i \in I$ as \begin{equation} (\pi^\ast b_2)_{|_I}= x_1^{N^0_{i_1}} y_1^{N^0_{i_2}} u_1(x_1, y_1) \quad {\rm and } \quad (\pi^\ast b_3)_{|_I}= x_1^{N^\infty_{i_1}} y_1^{N^\infty_{i_2}} u_1(x_1, y_1) @@ -652,7 +662,7 @@ in the case $I= \{i\}$ and $$\begin{cases} z^3 + x^{N^\infty_i}y^{N^\infty_j}, & \mathcal{M}(E_i), \mathcal{M}(E_j) \ne 1, \infty \\ -z^3 + x^{\frac{N^0_i}{2}}y^{\frac{N^0_j}{2}}z^2 + x^{N_i^\infty + M_i^\Delta}y^{N_j^\infty + M_ij^\Delta} , & \mathcal{M}(E_i) = \mathcal{M}(E_j) = 1 \\ +z^3 + x^{\frac{N^0_i}{2}}y^{\frac{N^0_j}{2}}z^2 + x^{N_i^\infty + M_i^\Delta}y^{N_j^\infty + M_j^\Delta} , & \mathcal{M}(E_i) = \mathcal{M}(E_j) = 1 \\ z^3 + x^{\frac{N^0_i}{2}}y^{\frac{N^0_j}{2}} z^2 + x^{N_i^\infty + M_i^\Delta}y^{N_j^\infty} , & \mathcal{M}(E_i) = 1, \mathcal{M}(E_j) = \mathbb{P}^1 \\ z^3 + x^{N^0_{i}}y^{N^0_{j}} z + x^{N^\infty_{i}}y^{N^\infty_{j}}, & {\rm else}\\ \end{cases}$$ @@ -988,9 +998,9 @@ Therefore, $F$ is non degenerate with respect to its Newton polyhedron if and on \section{Contibutions to the topological zeta function and the A´Campo monodromy zeta function of the trinomial forms} - \begin{proposition} -The monodromy zeta function and the local topological zeta function $Z(F,s)_0$ of $F$ equal -$$(1 - t^2)^{\alpha} \prod_{I} \zeta_{f_I}(t)^{-\chi(E_I^\circ)}$$ + \begin{proposition}\label{formestrpi} +The monodromy zeta function $\zeta_F(t)$ and the local topological zeta function $Z(F,s)_0$ of $F$ equal +$$(1 - t^3)^{\alpha} \prod_{I} \zeta_{f_I}(t)^{-\chi(E_I^\circ)}$$ $$\sum_I \chi(E^\circ_I) Z(f_I, \omega_I)_{o_I}$$ where $I$ varies over the statification associated to the resolution $\pi$ of $b_2b_3D_zF$, the exponent $\alpha$ is the number of points of blow-ups needed to obtain $\pi$, and $o_I$ is a point in $E^\circ_I$. @@ -998,9 +1008,12 @@ $$\sum_I \chi(E^\circ_I) Z(f_I, \omega_I)_{o_I}$$ \end{proposition} -\begin{example} - -\end{example} +\begin{example} The resolution $\pi$ of Example \ref{extriplepole} consist of $\alpha=15$ blow-ups and the associated stratification has 35 strata. According to Proposition \ref{formestrpi} we have +$$\zeta_F(t) = \frac{(1-t^3)^{15} \zeta^{}_{f_{2}} \zeta^{2}_{f_{4}} \zeta^{2}_{f_{5}} \zeta^{2}_{f_{6}} \zeta^{}_{f_{8}} \zeta^{}_{f_{10}} \zeta^{2}_{f_{12}} \zeta^{2}_{f_{13}} \zeta^{2}_{f_{14}} \zeta^{}_{f_{16}} } +{\zeta^{13}_{f_{1}} \zeta^{}_{f_{3}} \zeta^{4}_{f_{7}} \zeta^{}_{f_{9}} \zeta^{}_{f_{11}} \zeta^{4}_{f_{15}} \zeta^{}_{f_{17}} }$$ +$$Z(F,s)_0=-11Z(f_1,s)_0 + \sum_{i=2,8,10,16}Z(f_i,s)_0 - \sum_{i=3, 9,11,17} Z(f_i,s)_0 - 2 \sum_{i=7,15}Z(f_i,s)_0 + \sum_{(i,j) \in S} Z(f_{i,j},s)_0,$$ +where $S$ is the subset indexing the 0-dimensional strata of Example \ref{extriplepole} and we have omitted the symbols of differential forms associated to each strata for shortness sake. +\end{example} @@ -1208,7 +1221,27 @@ $$\frac{1}{\nu_1\nu_2(3s+1)} - \frac{2}{(3m_2s + 2\nu_1+m_2)(3n_2s + 2\nu_2+m_2) \subsection{} Third case $f= z^3 - x^{m_1}y^{n_1}z^2 - x^{m_3}y^{n_3}$ and such that $3m_1 \leq m_3, 3 n_1 \leq n_3$ and $3(m_1, n_1) \ne 2(m_3, n_3)$. - + \section{Reductions} + + \subsection{First reduction: Elimination of the 0-dimensional strata due to the strict transforms of $b_2$ and of $b_3$} The following Proposition shows that the o-dimensional strata associated to the strict transforms of $b_2$ and $b_3$ can eliminated from the computations of \ref{}. Each of these points can be added to the 1-dimensional strata satisfying that its clousere contains it, and the invariant is conputed with the trinomial form of the 1-dimensional strata. + + \begin{proposition}\label{red1}Assume that $E_i$ is an exceptional divisor of the resolution $\pi$ with $\pi(E_i)=0$ (resp. $\pi(E_i)=\infty$) and that $F^0_j$ is an strict transform of $b_2$ (resp. $b_3$) such that $E_i \cap F_j \ne \emptyset$. Then + $$Z(F_i, \omega_i)_{o_i}=Z(F_{i,j}, \omega_{i,j})_{o_{i,j}} $$ + \end{proposition} + +\begin{example} The proposition \ref{red1} can be applied to the strata +$$E^\circ_{9,32}=E_9 \cap F^0_{32}, \quad E^\circ_{17,33}=E_{17} \cap F^0_{33}, \quad E^\circ_{7,34}= E_7 \cap F^\infty_{34} \quad \hbox{ and } \quad E^\circ_{15,35}=E_{15} \cap F^\infty_{35}$$ + of Example \ref{extriplepole}. For instance, ifrom the formulas given in the previous section; we have +$$Z(F_9, \omega_9)_{o_9}=Z(F_{9,31}, \omega_{9,31})_{o_{9,31}}$$ +$$ Z(F_{15}, \omega_{15})_{o_{15}}=Z(F_{15,34}, \omega_{15,34})_{o_{15,34}}$$ +\end{example} + + \subsection{Second reduction: Elimination of connected components of exceptional divisors containing a $(-1)$-curve} + + After performing the First reduction, connected components of exceptional divisors + + \begin{proposition}\label{red2} + \end{proposition} \section{On generic projections}