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MASAKAZU SUZUKI

Proof. - We have ${\mathit{q}}_{\mathrm{1}}{\mathit{\delta }}_{\mathrm{1}}\mathrm{=}\mathit{p}{\mathit{\delta }}_{\mathrm{0}}$ by the corollary to Proposition 5.

Therefore, it is sufficient to prove (2) for $\mathit{k}\mathrm{\ge }\mathrm{2}$. Set $\mathit{\sigma }\mathrm{=}{\mathrm{i}}_{\mathit{k}}$, and let us

consider the surface ${\mathit{M}}_{\mathit{\sigma }}$ obtained by the $\mathrm{(}\mathit{\sigma }\mathrm{-}\mathrm{1}\mathrm{)}$-th blowing up in the

process to get $\mathit{M}$ from ${\mathit{M}}_{\mathrm{1}}$. We may say that ${\mathit{M}}_{\mathit{\sigma }}$ is the surface obtained

by the blowing down of ${\mathit{L}}_{\mathit{h}\mathrm{+}\mathrm{1}}\mathrm{,}$ ${\mathit{L}}_{\mathit{h}}$, ..., ${\mathit{L}}_{\mathit{k}\mathrm{+}\mathrm{1}}$ successively from $\mathit{M}$. Let

${\mathit{\pi }}_{\mathit{\sigma }}$ : $\mathit{M}\mathrm{\to }{\mathit{M}}_{\mathit{\sigma }}$ be the contraction mapping. As in the previous sections,

let us denote the proper images of $\overline{\mathit{C}}\mathrm{,}$ ${\overline{\mathit{C}}}_{\mathit{k}}\mathrm{,}$ ${\mathit{E}}_{\mathit{i}}$ in ${\mathit{M}}_{\mathit{\sigma }}$ by ${\overline{\mathit{C}}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}\mathrm{,}$ ${\overline{\mathit{C}}}_{\mathit{k}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}\mathrm{,}$ ${\mathit{E}}_{\mathit{i}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}$

respectively. By Theorem 3, ${\overline{\mathit{C}}}_{\mathit{k}\mathrm{+}\mathrm{1}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}$ intersects transversely ${\mathit{E}}_{\mathit{\sigma }}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}$ at the same

point $\mathit{Q}\mathrm{=}{\mathit{\pi }}_{\mathit{\sigma }}\mathrm{(}{\mathit{L}}_{\mathit{k}\mathrm{+}\mathrm{1}}\mathrm{\cup }\mathrm{\text{...}}\mathrm{\cup }{\mathit{L}}_{\mathit{h}\mathrm{+}\mathrm{1}}\mathrm{)}$ as ${\overline{\mathit{C}}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}$. lEience, the functions $\mathit{f}$ and $\mathit{g}\mathit{k}\mathrm{+}\mathrm{1}$

on ${\mathit{M}}_{\mathit{\sigma }}$ have the same indetermination point $\mathit{Q}\mathrm{\in }{\mathit{E}}_{\mathit{\sigma }}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}$. Let

${\mathit{P}}_{\mathit{f}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}\mathrm{=}{\displaystyle \sum }_{\mathit{i}\mathrm{=}\mathrm{0}}^{\mathit{\sigma }}{\mathit{\nu }}_{\mathit{i}}{\mathit{E}}_{\mathit{i}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}$, ${\mathit{P}}_{\mathit{g}\mathit{k}\mathrm{+}\mathrm{1}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}\mathrm{=}{\displaystyle \sum }_{\mathit{i}\mathrm{=}\mathrm{0}}^{\mathit{\sigma }}{\overline{\mathit{\nu }}}_{\mathit{i}}{\mathit{E}}_{\mathit{i}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}$

be the pole divisor of $\mathit{f}$ and $\mathit{g}\mathit{k}\mathrm{+}\mathrm{1}$ on ${\mathit{M}}_{\mathit{\sigma }}$ respectively. Let $\overline{\mathit{\delta }}\mathrm{0}\mathrm{,}{\overline{\mathit{\delta }}}_{\mathrm{1}}\mathrm{,}$ $\mathrm{\text{...}}\mathrm{,}{\overline{\mathit{\delta }}}_{\mathit{k}}$ be

the order of the pole of ${\mathit{g}}_{\mathit{k}\mathrm{+}\mathrm{1}}$ on ${\mathit{E}}_{\mathit{j}\mathrm{o}}\mathrm{(}\mathrm{=}{\mathit{E}}_{\mathrm{0}}\mathrm{)}\mathrm{,}$ ${\mathit{E}}_{{\mathit{j}}_{\mathrm{1}}}\mathrm{(}\mathrm{=}{\mathit{E}}_{\mathrm{1}}\mathrm{)}\mathrm{,}$ $\mathrm{\text{...}}\mathrm{,}$ ${\mathit{E}}_{{\mathit{j}}_{\mathit{k}}}$. We have

${\overline{\mathit{\delta }}}_{\mathrm{0}}\mathrm{=}{\overline{\mathit{\nu }}}_{{\mathit{j}}_{\mathrm{0}}}\mathrm{,}{\overline{\mathit{\delta }}}_{\mathrm{1}}\mathrm{=}{\overline{\mathit{\iota }\mathrm{/}}}_{{\mathit{j}}_{\mathrm{1}}}\mathrm{,}$ $\mathrm{\text{...}}\mathrm{,}{\overline{\mathit{\delta }}}_{\mathit{k}}\mathrm{=}{\overline{\mathit{\nu }}}_{{\mathit{j}}_{\mathit{k}}}$. The coefficients ${\mathit{\nu }}_{\mathit{i}}\mathrm{,}{\overline{\mathit{\nu }}}_{\mathit{i}}\mathrm{(}\mathit{i}\mathrm{=}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{}\mathrm{\text{...}}\mathrm{,}\mathrm{}\mathit{\sigma }\mathrm{)}$ are

the solutions of the following equations:

${\displaystyle \sum }_{\mathit{j}\mathrm{=}\mathrm{0}}^{\mathit{\sigma }}\mathrm{(}{\mathit{E}}_{\mathit{i}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}\mathrm{}\mathrm{.}\mathrm{}{\mathit{E}}_{\mathit{j}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}\mathrm{)}{\mathit{\nu }}_{\mathit{j}}\mathrm{=}\mathrm{\{}$

0 $\mathrm{(}\mathit{i}\mathrm{\ne }\mathit{\sigma }\mathrm{)}$

${\mathit{d}}_{\mathit{k}\mathrm{+}\mathrm{1}}$ $\mathrm{(}\mathit{i}\mathrm{=}\mathit{\sigma }\mathrm{)}$,

${\displaystyle \sum }_{\mathit{j}\mathrm{=}\mathrm{0}}^{\mathit{\sigma }}\mathrm{(}{\mathit{E}}_{\mathit{i}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}\mathrm{ċ}{\mathit{E}}_{\mathit{j}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}\mathrm{)}{\overline{\mathit{\nu }}}_{\mathit{j}}\mathrm{=}\mathrm{\{}$

0 $\mathrm{(}\mathit{i}\mathrm{\ne }\mathit{\sigma }\mathrm{)}$

1 $\mathrm{(}\mathit{i}\mathrm{=}\mathit{\sigma }\mathrm{)}$.

Hence, by Lemma 4, we have ${\mathit{\nu }}_{\mathit{i}}\mathrm{=}{\mathit{d}}_{\mathit{k}\mathrm{†}{\mathrm{1}}^{\overline{{U}}}\mathit{i}}$ for all $\mathrm{i}\mathrm{=}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{,}$ $\mathrm{\text{...}}\mathrm{,}$ $\mathit{\sigma }$. In

particular,

${\mathit{\delta }}_{\mathit{i}}\mathrm{=}{\overline{\mathit{\delta }}}_{\mathit{i}}\mathrm{ċ}{\mathit{d}}_{\mathit{k}\mathrm{+}\mathrm{1}}$, $\mathrm{(}\mathit{i}\mathrm{=}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{}\mathrm{\text{...}}\mathrm{,}\mathrm{}\mathit{k}\mathrm{)}$.

Therefore, in order to prove (2), it is sufficient to prove

(3) ${\mathit{q}}_{\mathit{k}}{\overline{\mathit{\delta }}}_{\mathit{k}}\mathrm{\in }\mathbb{N}{\overline{\mathit{\delta }}}_{\mathrm{0}}\mathrm{+}\mathbb{N}{\overline{\mathit{\delta }}}_{\mathrm{1}}\mathrm{+}\mathrm{\text{...}}\mathrm{+}\mathbb{N}{\overline{\mathit{\delta }}}_{\mathit{k}\mathrm{-}\mathrm{1}}$.

By Theorem 3, ${\overline{\mathit{C}}}_{\mathit{k}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}$ intersects ${\mathit{E}}_{{\mathit{j}}_{\mathit{k}}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}$ transversely and does not inter-

sects other components ${\mathit{E}}_{\mathit{i}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}\mathrm{(}\mathrm{i}\mathrm{\ne }{\mathit{j}}_{\mathit{k}}\mathrm{)}$. We have

${\overline{\mathit{\delta }}}_{\mathit{k}}\mathrm{=}\mathrm{(}{\mathit{P}}_{\mathit{9}\mathit{k}\mathrm{+}\mathrm{1}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}\mathrm{}\mathrm{.}\mathrm{}{\overline{\mathit{C}}}_{\mathit{k}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}\mathrm{)}$

$\mathrm{=}\mathrm{(}{\overline{\mathit{C}}}_{\mathit{k}\mathrm{+}\mathrm{1}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}\mathrm{}\mathrm{.}\mathrm{}{\overline{\mathit{C}}}_{\mathit{k}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}\mathrm{)}$

$\mathrm{=}$ $\mathrm{(}{\overline{\mathit{C}}}_{\mathit{k}\mathrm{+}\mathrm{1}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}\mathrm{}\mathrm{.}\mathrm{}{\mathit{P}}_{\mathit{9}\mathit{k}}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}\mathrm{)}$.