diff --git a/Mult3.pdf b/Mult3.pdf index cb073f2..f02383a 100644 Binary files a/Mult3.pdf and b/Mult3.pdf differ diff --git a/Mult3.tex b/Mult3.tex index f049320..43b4554 100644 --- a/Mult3.tex +++ b/Mult3.tex @@ -415,7 +415,7 @@ where $\omega = dx \wedge dy$ is the standard volume form of $\CC^2$. \begin{proposition} Let $\pi': X' \rightarrow \CC^2$ be the minimal resolution of $D_zF$. Then, $\pi'$ is a resolution of the meromorphic function $\mathcal{M} : \CC^2 \rightarrow \mathbb{P}^1$ defined by $\frac{b_2^3}{b_3^2}$ too. \end{proposition} -There are four possibilities for the behaviour of the numbers $N^0_j, N^\infty_j$ and $N^\Delta_j$ associated with an irreducible component $E_j$ of $(\pi \circ G)^{-1} \{G=0\}$ +There are four possibilities for the behaviour of the numbers $N^0_j, N^\infty_j$ and $N^\Delta_j$ associated with an irreducible component $E_j$ of $(\pi)^{-1} \{G=0\}$ \begin{itemize} \item $3 N^0_j > 2N^\infty_j = N^\Delta_j$ and it is called a \emph{zero} because in this case $\mathcal{M}(E_j)=0$; \item $2 N^\infty_j > 3N^0_j = N^\Delta_j$ and it is called a \emph{pole} because in this case $\mathcal{M}(E_j)=\infty$; @@ -586,7 +586,7 @@ N^\Delta_i& 25& {\bf 26}& {\bf 52}& {\bf 78}& {\bf 104}& {\bf 130}& {\bf 156}& { \hline \nu_i&2&3&5&7&9&11&13&6&11\\ \chi()&-11&1&-1&0&0&0&-2&1&-1\\ -E_i^2& -7& -2& -2& -2& -2& -2& -1&-2&-1\\ +E_i^2& -13& -2& -4& -2& -2& -2& -1&-2&-1\\ \hline \end{array}$$ @@ -645,7 +645,7 @@ If the strata $E^\circ_{i,j} \ne \emptyset$ and $\mathcal{M}(E_i)=1$ then either $$x_1= x_2, \quad y_1= y_2, \quad z = z_1 - x_2^\frac{N^0_i}{2} y_2^\frac{N^0_j}{2} + \cdots$$ such that (\ref{quasi-trinomial}) becomes either -$$z_1^3 + x_2^{\frac{N^0_i}{2}} y_2^{\frac{N^0_j}{2}} u'_1(x_2,y_2) z_1^2 + x_2^{N_i^\infty + M_i^\Delta} y_2^{N_i^\infty + M_i^\Delta} u'_2(x_2,y_2)$$ +$$z_1^3 + x_2^{\frac{N^0_i}{2}} y_2^{\frac{N^0_j}{2}} u'_1(x_2,y_2) z_1^2 + x_2^{N_i^\infty + M_i^\Delta} y_2^{N_j^\infty + M_j^\Delta} u'_2(x_2,y_2)$$ if both $\mathcal{M}(E_i) = \mathcal{M}(E_j)=1$ or $$z_1^3 + x_2^{\frac{N^0_i}{2}}y_2^{\frac{N^0_j}{2}} u'_1(x_2,y_2) z_1^2 + x_2^{N_i^\infty + M_i^\Delta}y_2^{N_j^\infty } u'_2(x_2,y_2)$$ @@ -656,7 +656,7 @@ if $\mathcal{M}(E_i)=1$ and $E_j$ is dicritical $$\begin{cases} z^3 - x^{N^\infty_{i}}, & \mathcal{M}(E_i) \ne 1, \infty \\ z^3 - x^{N^0_{i}} z + x^{N^\infty_{i}}, & \mathcal{M}(E_i) = \infty \\ -z^3 - x_2^{\frac{N^0_i}{2}} + x^{N_i^\infty + M_i^\Delta} , & \mathcal{M}(E_i) = 1, \\ +z^3 - x_2^{\frac{N^0_i}{2}}z^2 + x^{N_i^\infty + M_i^\Delta} , & \mathcal{M}(E_i) = 1, \\ \end{cases}$$ in the case $I= \{i\}$ and @@ -670,6 +670,47 @@ in the case $I= \{i, j\}$. \end{definition} +\begin{example} The following tables show all the trinomial forms and differential forms for the all the 0-dimensional strata and all the 1-dimensional strata with not zero Euler characteristic of Example \ref{extriplepole} + + +$$\begin{array}{c|c|c|c} +i&f_i&\omega_i& \chi()\\ +\hline +1& z^3 + x^4z^2 + x^{13}& (2, 1, 1)& -13\\ +2,10& z^3 + x^{10}z + x^{13}& (3, 1, 1)&1\\ +3,11& z^3 + x^{20}z + x^{26}& (5, 1, 1)& -1\\ +7,15& z^3 + \color{red}{x^{52}z} + x^{78}& (13, 1, 1)& -2\\ +8, 16& z^3 + x^{21}z + x^{26}& (6, 1, 1)& 1\\ + 9,17& z^3 + x^{42}z + x^{52}& (11, 1, 1)& -1\\ +\end{array}$$ + + + +$$\begin{array}{c|c|c} +(i,j)&f_{i,j}&\omega_{i,j}\\ +\hline +&&\\ +(1, 7), (1, 15)& z^3 + x^4y^{26}z^2 + x^{13}y^{78}& (2, 13, 1)\\ +(2,3), (10, 11)& z^3 + x^{10}y^{20}z + x^{13}y^{26}& (3, 5, 1)\\ +(3, 9), (11, 17)& z^3 + x^{20}y^{42}z + x^{26}y^{52}& (5, 11, 1)\\ +(3, 4), (11,12)& z^3 + x^{20}y^{28}z + x^{26}y^{39}& (5, 7, 1)\\ +(8, 9), (16,17)& z^3 + x^{21}y^{42}z + x^{26}y^{52}& (6, 11, 1)\\ +(4, 5), (12,13)& z^3 + x^{28}y^{36}z + x^{39}y^{52}& (7, 9, 1)\\ +(5, 6), (13,14)& z^3 + x^{36}y^{44}z + x^{52}y^{65}& (9, 11, 1)\\ +(6, 7), (14,15)& z^3 + x^{44}y^{52}z + x^{65}y^{78}& (11, 13, 1)\\ +(1, i), i=16, \dots, 28& z^3 + x^4z^2 + x^{13}y& (2, 1, 1)\\ +(9, 31), (17, 32)& z^3 + x^{42}yz + x^{52}& (11, 1, 1)\\ +(7, 33), (15, 34)& z^3 + x^{52}z + x^{78}y& (13, 1, 1)\\ +(7, 29) (15, 30)& z^3 + x^{26}z^2 + x^{78}y& (13, 1, 1)\\ +%(12, 'f')& z^3 + x^{42}yz + x^{52}& (11, 1, 1)\\ +%(16, 'g')& z^3 + x^{52}z + x^{78}y& (13, 1, 1)\\ +%(16, 'd')& z^3 + x^{26}z^2 + x^{78}y& (13, 1, 1) +\end{array}$$ + + +\end{example} + + \section{The quasi-trinomial case}\label{3nomial} @@ -1003,15 +1044,18 @@ The monodromy zeta function $\zeta_F(t)$ and the local topological zeta functio $$(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}$$ where $I$ varies over the statification associated to the resolution $\pi$ of $b_2b_3D_zF$, the exponent $\alpha$ is the number of points of blow-ups needed to obtain $\pi$, and $o_I$ is a point in $E^\circ_I$. + \end{proposition} - - \end{proposition} + \begin{proof} It follows from the the formula for $\zeta_f$ in terms of a partial resolution \cite[Theorem 1]{GSLM97} and the Stratum principle for topological zeta functions \cite[Stratum Principle 1.2]{ABCNLM02}. Each of the $\alpha$ factors of the form $(1-t^3)$ of $\zeta_f$ comes from the intersection of Milnor fiber of $F$ at $o$ and the line thru a center of blow-up of $\pi$. Each of these intersection sets consists of 3 points that are cyclically permuted by the monodromy. + \end{proof} \begin{example} The resolution $\pi$ of Example \ref{extriplepole} consist of $\alpha=15$ blow-ups and the associated stratification has 35 strata. According to Proposition \ref{formestrpi} we have $$\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,$$ +{\zeta^{13}_{f_{1}} \zeta^{}_{f_{3}} \zeta^{4}_{f_{7}} \zeta^{}_{f_{9}} \zeta^{}_{f_{11}} \zeta^{4}_{f_{15}} \zeta^{}_{f_{17}} }= \Phi^3_1 \Phi^{15}_2 \Phi_3 \Phi^2_6 \Phi^2_{13} \Phi^{15}_{26} \Phi^2_{78}$$ +where $\Phi_i$ is $i$-th the cyclotomic polymonial, and $Z(F,s)_0$ equals +$$-13Z(f_1,s)_0 + \sum_{i=2,8,10,16}Z(f_i,s)_0 - \sum_{i=3, 9,11,17} Z(f_i,s)_0 - 2 \sum_{i=7,15}Z(f_i,s)_0 + \sum_{(i,j) \in S} Z(f_{i,j},s)_0=$$ +$$\frac{-1/26s^3 + 2749/6084s^2 + 1969/4056s + 1/8}{(s+1)(s+1/2)^3}$$ 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} @@ -1223,25 +1267,37 @@ $$\frac{1}{\nu_1\nu_2(3s+1)} - \frac{2}{(3m_2s + 2\nu_1+m_2)(3n_2s + 2\nu_2+m_2) \section{Reductions} - \subsection{First reduction: Elimination of the 0-dimensional strata due to the strict transforms of $b_2$ and of $b_3$} The following Proposition shows that the o-dimensional strata associated to the strict transforms of $b_2$ and $b_3$ can eliminated from the computations of \ref{}. Each of these points can be added to the 1-dimensional strata satisfying that its clousere contains it, and the invariant is conputed with the trinomial form of the 1-dimensional strata. + \subsection{First reduction: Elimination of the 0-dimensional strata due to the strict transforms of $b_2$ and of $b_3$} The following Proposition shows that the o-dimensional strata associated to the strict transforms of $b_2$ and $b_3$ can eliminated. Each of these points can be added to the 1-dimensional strata satisfying that its clousure contains it, and the invariant is computed with the trinomial form of the 1-dimensional strata. \begin{proposition}\label{red1}Assume that $E_i$ is an exceptional divisor of the resolution $\pi$ with $\pi(E_i)=0$ (resp. $\pi(E_i)=\infty$) and that $F^0_j$ is an strict transform of $b_2$ (resp. $b_3$) such that $E_i \cap F_j \ne \emptyset$. Then $$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}}$$ +$$E^\circ_{9,31}=E_9 \cap F^0_{31}, \quad E^\circ_{17,32}=E_{17} \cap F^0_{32}, \quad E^\circ_{7,33}= E_7 \cap F^\infty_{33} \quad \hbox{ and } \quad E^\circ_{15,34}=E_{15} \cap F^\infty_{34}$$ + of Example \ref{extriplepole}. + + For the strata due to the strict transforms $F^0_{31}$ and $F^0_{32}$ of $b_2$, note that $\mathcal{M}(E_9)= \mathcal{M}(E_{17})=0$. The corresponding trinomial forms and differential forms are + $$f_9=f_{17}=f_{9,31}=f_{17,32}=z^3 - x^{52}, \qquad \omega_9=\omega_{17}=\omega_{9,31}=\omega_{17,32}= x^{11}yzdxdydz,$$ +From the formulas given in the previous section; we have +$$Z(F_9, \omega_9)_{o_9}=Z(F_{9,31}, \omega_{9,31})_{o_{9,31}}= Z(F_{17}, \omega_{17})_{o_{15}}=Z(F_{17,32}, \omega_{17,32})_{o_{17,32}} = \frac{37/858s + 85/1716}{s^2 +241/156 s + 85/156}.$$ + + For the strata due to the strict transforms $F^\infty_{33}$ and $F^\infty_{34}$ of $b_3$, note that $\mathcal{M}(E_7)= \mathcal{M}(E_{15})=0$. The corresponding trinomial forms and differential forms are + $$f_7=f_{15}=z^3 - x^{78}, \quad f_{7,33}=f_{15,34}=z^3 - x^{52}z + x^{78}y,$$ + and $$\omega_7=\omega_{15}=\omega_{7,33}=\omega_{15,34}= x^{13}yzdxdydz,$$ +From the formulas given in the previous section; we have +$$Z(F_7, \omega_7)_{o_7}=Z(F_{7,33}, \omega_{7,33})_{o_{7,33}}= Z(F_{15}, \omega_{15})_{o_{15}}=Z(F_{15,34}, \omega_{15,34})_{o_{15,34}} = \frac{1}{26(s^2 +3/2 s + 1/2}$$ \end{example} - \subsection{Second reduction: Elimination of connected components of exceptional divisors containing a $(-1)$-curve} + \subsection{Second reduction: Recursive elimination of exceptional $(-1)$-curve not intersecting the strict transform of the discrimiant} - After performing the First reduction, connected components of exceptional divisors + 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.. \begin{proposition}\label{red2} \end{proposition} + + + \subsection{Third reduction: $\QQ$-resolution of the discriminant} \section{On generic projections}