%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% This is an amstex file.                                       %%
%% It begins with the MRL style file.                            %%
%% Search for 'END OF STYLE FILE' to get to the paper itself.    %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%< %%\def\PSAMSFonts{TT}%  Blue Sky PS AMS fonts: True
%< \def\PSAMSFonts{TF}% Blue Sky PS AMS fonts: False
%---
%> \def\PSAMSFonts{TT}%  Blue Sky PS AMS fonts: True
%> %%\def\PSAMSFonts{TF}% Blue Sky PS AMS fonts: False

\input amstex
\def\filename{mrlt.sty}
\def\fileversion{2.1d}
\def\filedate{2-Sep-1999}
\expandafter\ifx\csname amsppt.sty\endcsname\endinput
  \expandafter\def\csname amsppt.sty\endcsname{2.1 (1-JUL-1991)}\fi
\xdef\fileversiontest{\fileversion\space(\filedate)}
\expandafter\ifx\csname\filename\endcsname\fileversiontest
  \message{[already loaded]}\endinput\fi
\expandafter\ifx\csname\filename\endcsname\relax % file not yet loaded
  \else\errmessage{Discrepancy in `\filename' file versions:
     version \csname\filename\endcsname\space already loaded, trying
     now to load version \fileversiontest}\fi
\expandafter\xdef\csname\filename\endcsname{%
  \catcode`\noexpand\@=\the\catcode`\@
  \expandafter\gdef\csname\filename\endcsname{%
     \fileversion\space(\filedate)}}
\catcode`\@=11
\message{version \fileversion\space(\filedate):}
\expandafter\ifx\csname styname\endcsname\relax
  \def\styname{AMSPPT}\def\styversion{2.1a}
\fi
\message{Loading utility definitions,}
\def\identity@#1{#1}
\def\nofrills@@#1{%
 \DN@{#1}%
 \ifx\next\nofrills \let\frills@\eat@
   \expandafter\expandafter\expandafter\next@\expandafter\eat@
  \else \let\frills@\identity@\expandafter\next@\fi}
\def\nofrillscheck#1{\def\nofrills@{\nofrills@@{#1}}%
  \futurelet\next\nofrills@}
\Invalid@\usualspace
\def\addto#1#2{\csname \expandafter\eat@\string#1@\endcsname
  \expandafter{\the\csname \expandafter\eat@\string#1@\endcsname#2}}
\newdimen\bigsize@
\def\big@#1#2{{\hbox{$\left#2\vcenter to#1\bigsize@{}%
  \right.\nulldelimiterspace\z@\m@th$}}}
\def\big{\big@\@ne}
\def\Big{\big@{1.5}}
\def\bigg{\big@\tw@}
\def\Bigg{\big@{2.5}}
\def\raggedcenter@{\leftskip\z@ plus.4\hsize \rightskip\leftskip
 \parfillskip\z@ \parindent\z@ \spaceskip.3333em \xspaceskip.5em
 \pretolerance9999\tolerance9999 \exhyphenpenalty\@M
 \hyphenpenalty\@M \let\\\linebreak}
\def\uppercasetext@#1{%
   {\spaceskip1.3\fontdimen2\the\font plus1.3\fontdimen3\the\font
    \def\ss{SS}\let\i=I\let\j=J\let\ae\AE\let\oe\OE
    \let\o\O\let\aa\AA\let\l\L
    \skipmath@#1$\skipmath@$}}
\def\skipmath@#1$#2${\uppercase{#1}%
  \ifx\skipmath@#2\else$#2$\expandafter\skipmath@\fi}
\def\add@missing#1{\expandafter\ifx\envir@end#1%
  \Err@{You seem to have a missing or misspelled
  \expandafter\string\envir@end ...}%
  \envir@end
\fi}
\newtoks\revert@
\def\envir@stack#1{\toks@\expandafter{\envir@end}%
  \edef\next@{\def\noexpand\envir@end{\the\toks@}%
    \revert@{\the\revert@}}%
  \revert@\expandafter{\next@}%
  \def\envir@end{#1}}
\begingroup
\catcode`\ =11
\gdef\revert@envir#1{\expandafter\ifx\envir@end#1%
\the\revert@%
\else\ifx\envir@end\enddocument \Err@{Extra \string#1}%
\else\expandafter\add@missing\envir@end\revert@envir#1%
\fi\fi}
\xdef\enddocument {\string\enddocument}%
\global\let\envir@end\enddocument %%%%%% don't remove the final space!
\endgroup\relax
\def\first@#1#2\end{#1}
\def\true@{TT}
\def\false@{TF}
\def\empty@{}
\begingroup  \catcode`\-=3
\long\gdef\notempty#1{%
  \expandafter\ifx\first@#1-\end-\empty@ \false@\else \true@\fi}
\endgroup
\message{more fonts,}
\def\PSAMSFonts{TT}%  Blue Sky PS AMS fonts: True
%%\def\PSAMSFonts{TF}% Blue Sky PS AMS fonts: False
\font@\tensmc=cmcsc10 \relax
\if\PSAMSFonts
  \font@\sevenex=cmex10 at 7pt
\else
  \font@\sevenex=cmex7 \relax
\fi
\font@\sevenit=cmti7 \relax
\font@\eightrm=cmr8 \relax % preloaded in plain.tex
\font@\sixrm=cmr6 \relax % preloaded in plain.tex
\font@\eighti=cmmi8 \relax     \skewchar\eighti='177 % preloaded
\font@\sixi=cmmi6 \relax       \skewchar\sixi='177   % preloaded
\font@\eightsy=cmsy8 \relax    \skewchar\eightsy='60 % preloaded
\font@\sixsy=cmsy6 \relax      \skewchar\sixsy='60   % preloaded
\if\PSAMSFonts
  \font@\eightex=cmex10 at 8pt
\else
  \font@\eightex=cmex8 \relax
\fi
\font@\eightbf=cmbx8 \relax % preloaded in plain.tex
\font@\sixbf=cmbx6 \relax   % preloaded in plain.tex
\font@\eightit=cmti8 \relax % preloaded in plain.tex
\font@\eightsl=cmsl8 \relax % preloaded in plain.tex
\if\PSAMSFonts
  \font@\eightsmc=cmcsc10 at 8pt
\else
  \font@\eightsmc=cmcsc8 \relax
\fi
\font@\eighttt=cmtt8 \relax % preloaded in plain.tex
%% Nine-point fonts are not needed but are included here, commented
%% out, to make it easier for a user to add them if they are needed.
%%\font@\ninerm=cmr9 \relax
%%\font@\ninei=cmmi9 \relax   \skewchar\ninei='177
%%\font@\ninesy=cmsy9 \relax  \skewchar\ninesy='60
%%\if\PSAMSFONTS
%%  \font@\nineex=cmex10 at9pt % non-AMSfonts substitute
%%\else
%%  \font@\nineex=cmex9 \relax
%%\fi
%%\font@\ninebf=cmbx9 \relax
%%\font@\nineit=cmti9 \relax
%%\font@\ninesl=cmsl9 \relax
%%\font@\ninesmc=cmcsc9 \relax
%%
%%\font@\ninemsa=msam9 \relax
%%\font@\ninemsb=msbm9 \relax
%%\font@\nineeufm=eufm9 \relax
%%     To use amsppt.sty without AMSFonts, comment out the following
%%     two lines (and refer to the lines above that begin with double
%%     percent signs); to load extra math symbols only on demand (with
%%     \newsymbol) comment out the second line.
\loadeufm \loadmsam \loadmsbm
\message{symbol names}\UseAMSsymbols\message{,}
\newtoks\tenpoint@
\def\tenpoint{\normalbaselineskip12\p@
 \abovedisplayskip12\p@ plus3\p@ minus9\p@
 \belowdisplayskip\abovedisplayskip
 \abovedisplayshortskip\z@ plus3\p@
 \belowdisplayshortskip7\p@ plus3\p@ minus4\p@
 \textonlyfont@\rm\tenrm \textonlyfont@\it\tenit
 \textonlyfont@\sl\tensl \textonlyfont@\bf\tenbf
 \textonlyfont@\smc\tensmc \textonlyfont@\tt\tentt
 \ifsyntax@ \def\big##1{{\hbox{$\left##1\right.$}}}%
  \let\Big\big \let\bigg\big \let\Bigg\big
 \else
   \textfont\z@\tenrm  \scriptfont\z@\sevenrm
       \scriptscriptfont\z@\fiverm
   \textfont\@ne\teni  \scriptfont\@ne\seveni
       \scriptscriptfont\@ne\fivei
   \textfont\tw@\tensy \scriptfont\tw@\sevensy
       \scriptscriptfont\tw@\fivesy
   \textfont\thr@@\tenex \scriptfont\thr@@\sevenex
        \scriptscriptfont\thr@@\sevenex
   \textfont\itfam\tenit \scriptfont\itfam\sevenit
        \scriptscriptfont\itfam\sevenit
   \textfont\bffam\tenbf \scriptfont\bffam\sevenbf
        \scriptscriptfont\bffam\fivebf
   \setbox\strutbox\hbox{\vrule height8.5\p@ depth3.5\p@ width\z@}%
   \setbox\strutbox@\hbox{\lower.5\normallineskiplimit\vbox{%
        \kern-\normallineskiplimit\copy\strutbox}}%
   \setbox\z@\vbox{\hbox{$($}\kern\z@}\bigsize@1.2\ht\z@
  \fi
  \normalbaselines\rm\dotsspace@1.5mu\ex@.2326ex\jot3\ex@
  \the\tenpoint@}
\newtoks\eightpoint@
\def\eightpoint{\normalbaselineskip10\p@
 \abovedisplayskip10\p@ plus2.4\p@ minus7.2\p@
 \belowdisplayskip\abovedisplayskip
 \abovedisplayshortskip\z@ plus2.4\p@
 \belowdisplayshortskip5.6\p@ plus2.4\p@ minus3.2\p@
 \textonlyfont@\rm\eightrm \textonlyfont@\it\eightit
 \textonlyfont@\sl\eightsl \textonlyfont@\bf\eightbf
 \textonlyfont@\smc\eightsmc \textonlyfont@\tt\eighttt
 \ifsyntax@\def\big##1{{\hbox{$\left##1\right.$}}}%
  \let\Big\big \let\bigg\big \let\Bigg\big
 \else
  \textfont\z@\eightrm \scriptfont\z@\sixrm
       \scriptscriptfont\z@\fiverm
  \textfont\@ne\eighti \scriptfont\@ne\sixi
       \scriptscriptfont\@ne\fivei
  \textfont\tw@\eightsy \scriptfont\tw@\sixsy
       \scriptscriptfont\tw@\fivesy
  \textfont\thr@@\eightex \scriptfont\thr@@\sevenex
   \scriptscriptfont\thr@@\sevenex
  \textfont\itfam\eightit \scriptfont\itfam\sevenit
   \scriptscriptfont\itfam\sevenit
  \textfont\bffam\eightbf \scriptfont\bffam\sixbf
   \scriptscriptfont\bffam\fivebf
 \setbox\strutbox\hbox{\vrule height7\p@ depth3\p@ width\z@}%
 \setbox\strutbox@\hbox{\raise.5\normallineskiplimit\vbox{%
   \kern-\normallineskiplimit\copy\strutbox}}%
 \setbox\z@\vbox{\hbox{$($}\kern\z@}\bigsize@1.2\ht\z@
 \fi
 \normalbaselines\eightrm\dotsspace@1.5mu\ex@.2326ex\jot3\ex@
 \the\eightpoint@}
\def\linespacing#1{%
  \addto\tenpoint{\normalbaselineskip=#1\normalbaselineskip
    \normalbaselines
    \setbox\strutbox=\hbox{\vrule height.7\normalbaselineskip
      depth.3\normalbaselineskip}%
    \setbox\strutbox@\hbox{\raise.5\normallineskiplimit
      \vbox{\kern-\normallineskiplimit\copy\strutbox}}%
  }%
  \addto\eightpoint{\normalbaselineskip=#1\normalbaselineskip
    \normalbaselines
    \setbox\strutbox=\hbox{\vrule height.7\normalbaselineskip
      depth.3\normalbaselineskip}%
    \setbox\strutbox@\hbox{\raise.5\normallineskiplimit
      \vbox{\kern-\normallineskiplimit\copy\strutbox}}%
  }%
}
\if\PSAMSFonts
  \def\extrafont@#1#2#3{%
    \font#1=#2%
      \ifnum#3=9 10 at9pt%
      \else\ifnum#3=8 10 at8pt%
      \else\ifnum#3=6 7 at6pt%
              \else #3\fi\fi\fi\relax}
\else
  \def\extrafont@#1#2#3{\font#1=#2#3\relax}
\fi
\def\loadextrasizes@#1#2#3#4#5#6#7{%
 \ifx\undefined#1%
 \else \extrafont@{#4}{#2}{8}\extrafont@{#6}{#2}{6}%
   \ifsyntax@
   \else
     \addto\tenpoint{\textfont#1#3\scriptfont#1#5%
       \scriptscriptfont#1#7}%
    \addto\eightpoint{\textfont#1#4\scriptfont#1#6%
       \scriptscriptfont#1#7}%
   \fi
 \fi
}
\def\loadextrafonts@{%
  \loadextrasizes@\msafam{msam}%
    \tenmsa\eightmsa\sevenmsa\sixmsa\fivemsa
  \loadextrasizes@\msbfam{msbm}%
    \tenmsb\eightmsb\sevenmsb\sixmsb\fivemsb
  \loadextrasizes@\eufmfam{eufm}%
    \teneufm\eighteufm\seveneufm\sixeufm\fiveeufm
  \loadextrasizes@\eufbfam{eufb}%
    \teneufb\eighteufb\seveneufb\sixeufb\fiveeufb
  \loadextrasizes@\eusmfam{eusm}%
    \teneusm\eighteusm\seveneusm\sixeusm\fiveeusm
  \loadextrasizes@\eusbfam{eusb}%
    \teneusb\eighteusb\seveneusb\sixeusb\fiveeusb
  \loadextrasizes@\eurmfam{eurm}%
    \teneurm\eighteurm\seveneurm\sixeurm\fiveeurm
  \loadextrasizes@\eurbfam{eurb}%
    \teneurb\eighteurb\seveneurb\sixeurb\fiveeurb
  \loadextrasizes@\cmmibfam{cmmib}%
    \tencmmib\eightcmmib\sevencmmib\sixcmmib\fivecmmib
  \loadextrasizes@\cmbsyfam{cmbsy}%
    \tencmbsy\eightcmbsy\sevencmbsy\sixcmbsy\fivecmbsy
  \let\loadextrafonts@\empty@
}
\message{page dimension settings,}
\parindent1pc
\newdimen\normalparindent \normalparindent\parindent
\normallineskiplimit\p@
\newdimen\indenti \indenti=2pc
\def\pageheight#1{\vsize#1\relax}
\def\pagewidth#1{\hsize#1%
   \captionwidth@\hsize \advance\captionwidth@-2\indenti}
\pagewidth{30pc} \pageheight{47pc}
\let\magnification=\mag
\message{top matter,}
\def\topmatter{\loadextrafonts@ \let\topmatter\relax}
\def\chapterno@{\uppercase\expandafter{\romannumeral\chaptercount@}}
\newcount\chaptercount@
\def\chapter{\let\savedef@\chapter
  \def\chapter##1{\let\chapter\savedef@
  \leavevmode\hskip-\leftskip
   \rlap{\vbox to\z@{\vss\centerline{\eightpoint
   \frills@{CHAPTER\space\afterassignment\chapterno@
       \global\chaptercount@=}%
   ##1\unskip}\baselineskip2pc\null}}\hskip\leftskip}%
 \nofrillscheck\chapter}
\newbox\titlebox@

\def\title{\let\savedef@\title
 \def\title##1\endtitle{
% {\noindent{\eightrm {\it Mathematical Research Letters~
%}{\eightbf
% \volno}, {\number\pageno}--\lpageno\ (\yrno)} \par\bigskip\bigskip}
   \let\title\savedef@
   \global\setbox\titlebox@\vtop{\tenpoint\bf
   \raggedcenter@
   \baselineskip1.3\baselineskip
   \frills@\uppercasetext@{##1}\endgraf}%
 \ifmonograph@ \edef\next{\the\leftheadtoks}%
    \ifx\next\empty@
    \leftheadtext{##1}\fi
 \fi
 \edef\next{\the\rightheadtoks}\ifx\next\empty@ \rightheadtext{##1}\fi
 }%
 \nofrillscheck\title}
%\def\title{\let\savedef@\title
% \def\title##1\endtitle{\let\title\savedef@
%   \global\setbox\titlebox@\vtop{\tenpoint\bf
%   \raggedcenter@
%   \baselineskip1.3\baselineskip
%   \frills@\uppercasetext@{##1}\endgraf}%
% \ifmonograph@ \edef\next{\the\leftheadtoks}%
%    \ifx\next\empty@
%    \leftheadtext{##1}\fi
% \fi
% \edef\next{\the\rightheadtoks}\ifx\next\empty@ \rightheadtext{##1}\fi
% }%
% \nofrillscheck\title}
\newbox\authorbox@
\def\author#1\endauthor{\global\setbox\authorbox@
 \vbox{\tenpoint\smc\raggedcenter@
 #1\endgraf}\relaxnext@ \edef\next{\the\leftheadtoks}%
 \ifx\next\empty@\leftheadtext{#1}\fi}
\newbox\affilbox@
\def\affil#1\endaffil{\global\setbox\affilbox@
 \vbox{\tenpoint\raggedcenter@#1\endgraf}}
\newcount\addresscount@
\addresscount@\z@
\def\address#1\endaddress{\global\advance\addresscount@\@ne
  \expandafter\gdef\csname address\number\addresscount@\endcsname
  {\nobreak\vskip12\p@ minus6\p@\indent\eightpoint\smc#1\par}}
\def\curraddr{\let\savedef@\curraddr
  \def\curraddr##1\endcurraddr{\let\curraddr\savedef@
  \toks@\expandafter\expandafter\expandafter{%
       \csname address\number\addresscount@\endcsname}%
  \toks@@{##1}%
  \expandafter\xdef\csname address\number\addresscount@\endcsname
  {\the\toks@\endgraf\noexpand\nobreak
    \indent{\noexpand\rm
    \frills@{{\noexpand\it Current address\noexpand\/}:\space}%
    \def\noexpand\usualspace{\space}\the\toks@@\unskip}}}%
  \nofrillscheck\curraddr}
\def\email{\let\savedef@\email
  \def\email##1\endemail{\let\email\savedef@
  \toks@{\def\usualspace{{\it\enspace}}\endgraf\indent\eightpoint}%
%  \toks@@{##1\par}%
 % v2.1c -
 % \toks@@{##1\par}%
   \toks@@{\tt ##1\par}%
  \expandafter\xdef\csname email\number\addresscount@\endcsname
  {\the\toks@\frills@{{\noexpand\it E-mail address\noexpand\/}:%
     \noexpand\enspace}\the\toks@@}}%
  \nofrillscheck\email}
\def\thedate@{}
\def\date#1\enddate{\gdef\thedate@{\tenpoint#1\unskip}}
\def\thethanks@{}
\def\thanks#1\endthanks{%
  \ifx\thethanks@\empty@ \gdef\thethanks@{\eightpoint#1}%
  \else
    \expandafter\gdef\expandafter\thethanks@\expandafter{%
     \thethanks@\endgraf#1}%
  \fi}
\def\thekeywords@{}
\def\keywords{\let\savedef@\keywords
  \def\keywords##1\endkeywords{\let\keywords\savedef@
  \toks@{\def\usualspace{{\it\enspace}}\eightpoint}%
  \toks@@{##1\unskip.}%
  \edef\thekeywords@{\the\toks@\frills@{{\noexpand\it
    Key words and phrases.\noexpand\enspace}}\the\toks@@}}%
 \nofrillscheck\keywords}
\def\thesubjclass@{}
\def\subjclass{\let\savedef@\subjclass
 \def\subjclass##1\endsubjclass{\let\subjclass\savedef@
   \toks@{\def\usualspace{{\rm\enspace}}\eightpoint}%
   \toks@@{##1\unskip.}%
   \edef\thesubjclass@{\the\toks@
     \frills@{{\noexpand\rm2000 {\noexpand\it Mathematics Subject
       Classification}.\noexpand\enspace}}%
     \the\toks@@}}%
  \nofrillscheck\subjclass}
\newbox\abstractbox@
\def\abstract{\let\savedef@\abstract
 \def\abstract{\let\abstract\savedef@
  \setbox\abstractbox@\vbox\bgroup\noindent$$\vbox\bgroup
  \def\envir@end{\endabstract}\advance\hsize-2\indenti
  \def\usualspace{\enspace}\eightpoint \noindent
  \frills@{{\smc Abstract.\enspace}}}%
 \nofrillscheck\abstract}
\def\endabstract{\par\unskip\egroup$$\egroup}
\def\widestnumber{\begingroup \let\head\relax\let\subhead\relax
  \let\subsubhead\relax \expandafter\endgroup\setwidest@}
\def\setwidest@#1#2{%
   \ifx#1\head\setbox\tocheadbox@\hbox{#2.\enspace}%
   \else\ifx#1\subhead\setbox\tocsubheadbox@\hbox{#2.\enspace}%
   \else\ifx#1\subsubhead\setbox\tocsubheadbox@\hbox{#2.\enspace}%
   \else\ifx#1\key\refstyle A%
       \setboxz@h{\refsfont@\keyformat{#2}}%
       \refindentwd\wd\z@
   \else\ifx#1\no\refstyle C%
       \setboxz@h{\refsfont@\keyformat{#2}}%
       \refindentwd\wd\z@
   \else\ifx#1\page\setbox\z@\hbox{\quad\bf#2}%
       \pagenumwd\wd\z@
   \else\ifx#1\item
       \setboxz@h{(#2)}\rosteritemwd\wdz@
   \else\message{\string\widestnumber\space not defined for this
      option (\string#1)}%
\fi\fi\fi\fi\fi\fi\fi}
\newif\ifmonograph@
\def\Monograph{\monograph@true \let\headmark\rightheadtext
  \let\varindent@\indent \def\headfont@{\bf}\def\proclaimheadfont@{\smc}%
  \def\remarkheadfont@{\smc}}
\let\varindent@\noindent
\newbox\tocheadbox@    \newbox\tocsubheadbox@
\newbox\tocbox@
\newdimen\pagenumwd
\def\toc{\toc@{Contents}}
\def\newtocdefs{%
   \def \title##1\endtitle
       {\penaltyandskip@\z@\smallskipamount
        \hangindent\wd\tocheadbox@\noindent{\bf##1}}%
   \def \chapter##1{%
        Chapter \uppercase\expandafter{%
              \romannumeral##1.\unskip}\enspace}%
   \def \specialhead##1\endspecialhead
       {\par\hangindent\wd\tocheadbox@ \noindent##1\par}%
   \def \head##1 ##2\endhead
       {\par\hangindent\wd\tocheadbox@ \noindent
        \if\notempty{##1}\hbox to\wd\tocheadbox@{\hfil##1\enspace}\fi
        ##2\par}%
   \def \subhead##1 ##2\endsubhead
       {\par\vskip-\parskip {\normalbaselines
        \advance\leftskip\wd\tocheadbox@
        \hangindent\wd\tocsubheadbox@ \noindent
        \if\notempty{##1}%
              \hbox to\wd\tocsubheadbox@{##1\unskip\hfil}\fi
         ##2\par}}%
   \def \subsubhead##1 ##2\endsubsubhead
       {\par\vskip-\parskip {\normalbaselines
        \advance\leftskip\wd\tocheadbox@
        \hangindent\wd\tocsubheadbox@ \noindent
        \if\notempty{##1}%
              \hbox to\wd\tocsubheadbox@{##1\unskip\hfil}\fi
        ##2\par}}}
\def\toc@#1{\relaxnext@
 \DN@{\ifx\next\nofrills\DN@\nofrills{\nextii@}%
      \else\DN@{\nextii@{{#1}}}\fi
      \next@}%
 \DNii@##1{%
\ifmonograph@\bgroup\else\setbox\tocbox@\vbox\bgroup
   \centerline{\headfont@\ignorespaces##1\unskip}\nobreak
   \vskip\belowheadskip \fi
   \def\page####1%
       {\unskip\penalty\z@\null\hfil
        \rlap{\hbox to\pagenumwd{\quad\hfil####1}}%
              \hfilneg\penalty\@M}%
   \setbox\tocheadbox@\hbox{0.\enspace}%
   \setbox\tocsubheadbox@\hbox{0.0.\enspace}%
   \leftskip\indenti \rightskip\leftskip
   \setboxz@h{\bf\quad000}\pagenumwd\wd\z@
   \advance\rightskip\pagenumwd
   \newtocdefs
 }%
 \FN@\next@}
\def\endtoc{\par\egroup}
\let\pretitle\relax
\let\preauthor\relax
\let\preaffil\relax
\let\predate\relax
\let\preabstract\relax
\let\prepaper\relax
\def\dedicatory #1\enddedicatory{\def\preabstract{{\medskip
  \eightpoint\it \raggedcenter@#1\endgraf}}}
\def\thetranslator@{}
\def\translator{%
  \let\savedef@\translator
  \def\translator##1\endtranslator{\let\translator\savedef@
    \edef\thetranslator@{\noexpand\nobreak\noexpand\medskip
      \noexpand\line{\noexpand\eightpoint\hfil
      \frills@{Translated by \uppercase}{##1}\qquad\qquad}%
       \noexpand\nobreak}}%
  \nofrillscheck\translator}
\outer\def\endtopmatter{\add@missing\endabstract
 \edef\next{\the\leftheadtoks}\ifx\next\empty@
  \expandafter\leftheadtext\expandafter{\the\rightheadtoks}\fi
 \ifmonograph@\else
   \ifx\thesubjclass@\empty@\else \makefootnote@{}{\thesubjclass@}\fi
   \ifx\thekeywords@\empty@\else \makefootnote@{}{\thekeywords@}\fi
   \ifx\thethanks@\empty@\else \makefootnote@{}{\thethanks@}\fi
 \fi
  \pretitle
  \begingroup % to localize variant topskip
  \ifmonograph@ \topskip7pc \else \topskip4pc \fi
  \box\titlebox@
  \endgroup
  \preauthor
  \ifvoid\authorbox@\else \vskip2.5pcplus1pc\unvbox\authorbox@\fi
  \preaffil
  \ifvoid\affilbox@\else \vskip1pcplus.5pc\unvbox\affilbox@\fi
  \predate
  \ifx\thedate@\empty@\else
       \vskip1pcplus.5pc\line{\hfil\thedate@\hfil}\fi
  \preabstract
  \ifvoid\abstractbox@\else
       \vskip1.5pcplus.5pc\unvbox\abstractbox@ \fi
  \ifvoid\tocbox@\else\vskip1.5pcplus.5pc\unvbox\tocbox@\fi
  \prepaper
  \vskip2pcplus1pc\relax
}
\def\document{%
  \loadextrafonts@
  \let\fontlist@\relax\let\alloclist@\relax
  \tenpoint}
\message{section heads,}
\newskip\aboveheadskip       \aboveheadskip\bigskipamount
\newdimen\belowheadskip      \belowheadskip6\p@
\def\headfont@{\bf}
\def\penaltyandskip@#1#2{\par\skip@#2\relax
  \ifdim\lastskip<\skip@\relax\removelastskip
      \ifnum#1=\z@\else\penalty@#1\relax\fi\vskip\skip@
  \else\ifnum#1=\z@\else\penalty@#1\relax\fi\fi}
\def\nobreak{\penalty\@M
  \ifvmode\gdef\penalty@{\global\let\penalty@\penalty\count@@@}%
  \everypar{\global\let\penalty@\penalty\everypar{}}\fi}
\let\penalty@\penalty
\def\heading#1\endheading{\head#1\endhead}
\def\subheading{\DN@{\ifx\next\nofrills
    \expandafter\subheading@
  \else \expandafter\subheading@\expandafter\empty@
  \fi}%
  \FN@\next@
}
\def\subheading@#1#2{\subhead#1#2\endsubhead}
\def\specialheadfont@{\bf}
\outer\def\specialhead{%
  \add@missing\endroster \add@missing\enddefinition
  \add@missing\enddemo \add@missing\endexample
  \add@missing\endproclaim
  \penaltyandskip@{-200}\aboveheadskip
  \begingroup\interlinepenalty\@M\rightskip\z@ plus\hsize
  \let\\\linebreak
  \specialheadfont@\noindent}
\def\endspecialhead{\par\endgroup\nobreak\vskip\belowheadskip}
\outer\def\head#1\endhead{%
  \add@missing\endroster \add@missing\enddefinition
  \add@missing\enddemo \add@missing\endexample
  \add@missing\endproclaim
  \penaltyandskip@{-200}\aboveheadskip
  {\headfont@\raggedcenter@\interlinepenalty\@M
  #1\endgraf}\headmark{#1}%
  \nobreak
  \vskip\belowheadskip}
\let\headmark\eat@
\def\restoredef@#1{\relax\let#1\savedef@\let\savedef@\relax}
\newskip\subheadskip       \subheadskip\medskipamount
\def\subheadfont@{\bf}
\outer\def\subhead{%
  \add@missing\endroster \add@missing\enddefinition
  \add@missing\enddemo \add@missing\endexample
  \add@missing\endproclaim
  \let\savedef@\subhead \let\subhead\relax
  \def\subhead##1\endsubhead{\restoredef@\subhead
    \penaltyandskip@{-100}\subheadskip
    \varindent@{\def\usualspace{{\subheadfont@\enspace}}%
    \subheadfont@\ignorespaces##1\unskip\frills@{.\enspace}}%
    \ignorespaces}%
  \nofrillscheck\subhead}
\newskip\subsubheadskip       \subsubheadskip\medskipamount
\def\subsubheadfont@{\it}
\outer\def\subsubhead{%
  \add@missing\endroster \add@missing\enddefinition
  \add@missing\enddemo
  \add@missing\endexample \add@missing\endproclaim
  \let\savedef@\subsubhead \let\subsubhead\relax
  \def\subsubhead##1\endsubsubhead{\restoredef@\subsubhead
    \penaltyandskip@{-50}\subsubheadskip
      {\def\usualspace{\/{\it\enspace}}%
    \subsubheadfont@##1\unskip\frills@{.\enspace}}}%
  \nofrillscheck\subsubhead}
\message{theorems/proofs/definitions/remarks,}
\def\proclaimheadfont@{\bf}
\def\proclaimfont{\it}
\outer\def\proclaim{%
  \let\savedef@\proclaim \let\proclaim\relax
  \add@missing\endroster \add@missing\enddefinition
  \add@missing\endproclaim \envir@stack\endproclaim
 \def\proclaim##1{\restoredef@\proclaim
   \penaltyandskip@{-100}\medskipamount\varindent@
   \def\usualspace{{\proclaimheadfont@\enspace}}\proclaimheadfont@
   \ignorespaces##1\unskip\frills@{.\enspace}%
  \proclaimfont\ignorespaces}%
 \nofrillscheck\proclaim}
\def\endproclaim{\revert@envir\endproclaim \par\rm
  \penaltyandskip@{55}\medskipamount}
\def\remarkheadfont@{\it}
\def\remark{\let\savedef@\remark \let\remark\relax
  \add@missing\endroster \add@missing\endproclaim
  \envir@stack\endremark
  \def\remark##1{\restoredef@\remark
    \penaltyandskip@\z@\medskipamount
  {\def\usualspace{{\remarkheadfont@\enspace}}%
  \varindent@\remarkheadfont@\ignorespaces##1\unskip%
  \frills@{.\enspace}}\rm
  \ignorespaces}\nofrillscheck\remark}
\def\endremark{\par\revert@envir\endremark}
\ifx\undefined\square
  \def\square{\vrule width.6em height.5em depth.1em\relax}\fi
\def\qed{\ifhmode\unskip\nobreak\fi\quad
  \ifmmode\square\else$\m@th\square$\fi}
\def\demo{\DN@{\ifx\next\nofrills
    \DN@####1####2{\remark####1{####2}\envir@stack\enddemo
      \ignorespaces}%
  \else
    \DN@####1{\remark{####1}\envir@stack\enddemo\ignorespaces}%
  \fi
  \next@}%
\FN@\next@}

\def\enddemo{\par\revert@envir\enddemo \endremark\medskip}
\def\definition{\let\savedef@\definition \let\definition\relax
  \add@missing\endproclaim \add@missing\endroster
  \add@missing\enddefinition \envir@stack\enddefinition
   \def\definition##1{\restoredef@\definition
     \penaltyandskip@{-100}\medskipamount
        {\def\usualspace{{\proclaimheadfont@\enspace}}%
        \varindent@\proclaimheadfont@\ignorespaces##1\unskip
        \frills@{.\proclaimheadfont@\enspace}}%
        \rm \ignorespaces}%
  \nofrillscheck\definition}
\def\enddefinition{\revert@envir\enddefinition
  \par\medskip}
\def\example{\DN@{\ifx\next\nofrills
    \DN@####1####2{\definition####1{####2}\envir@stack\endexample
      \ignorespaces}%
  \else
    \DN@####1{\definition{####1}\envir@stack\endexample\ignorespaces}%
  \fi
  \next@}%
\FN@\next@}
\def\endexample{\revert@envir\endexample \enddefinition }
\message{rosters,}
\newdimen\rosteritemwd
\rosteritemwd16pt % approximately the width of (iii) in 10 point text
\newcount\rostercount@
\newif\iffirstitem@
\let\plainitem@\item
\newtoks\everypartoks@
\def\par@{\everypartoks@\expandafter{\the\everypar}\everypar{}}
\def\leftskip@{}
\def\roster{%
  \envir@stack\endroster
 \edef\leftskip@{\leftskip\the\leftskip}%
 \relaxnext@
 \rostercount@\z@% Initialize \rostercount@ to 0.
 \def\item{\FN@\rosteritem@}%      \item, now redefined, has
 \DN@{\ifx\next\runinitem\let\next@\nextii@\else
  \let\next@\nextiii@\fi\next@}%
 \DNii@\runinitem% If \runinitem occurs, \nextii@ must kill it off.
  {\unskip% This unskips any space before the original \roster.
   \DN@{\ifx\next[\let\next@\nextii@\else
    \ifx\next"\let\next@\nextiii@\else\let\next@\nextiv@\fi\fi\next@}%
   \DNii@[####1]{\rostercount@####1\relax
    \enspace\therosteritem{\number\rostercount@}~\ignorespaces}%
   \def\nextiii@"####1"{\enspace{\rm####1}~\ignorespaces}%
   \def\nextiv@{\enspace\therosteritem1\rostercount@\@ne~}%
   \par@\firstitem@false% Before doing any of this we still change
   \FN@\next@}%      End of definition of \nextii@\runinitem.
 \def\nextiii@{\par\par@% End the present paragraph, change \everypar
  \penalty\@m\smallskip\vskip-\parskip
  \firstitem@true}
 \FN@\next@}
\def\rosteritem@{\iffirstitem@\firstitem@false
  \else\par\vskip-\parskip\fi
 \leftskip\rosteritemwd \advance\leftskip\normalparindent
 \advance\leftskip.5em \noindent
 \DNii@[##1]{\rostercount@##1\relax\itembox@}%
 \def\nextiii@"##1"{\def\therosteritem@{\rm##1}\itembox@}%
 \def\nextiv@{\advance\rostercount@\@ne\itembox@}%
 \def\therosteritem@{\therosteritem{\number\rostercount@}}%
 \ifx\next[\let\next@\nextii@\else\ifx\next"\let\next@\nextiii@\else
  \let\next@\nextiv@\fi\fi\next@}
\def\itembox@{\llap{\hbox to\rosteritemwd{\hss
  \kern\z@ % kern to thwart \unskip in \rom
  \therosteritem@}\enspace}\ignorespaces}
\def\therosteritem#1{\rom{(\ignorespaces#1\unskip)}}
\newif\ifnextRunin@
\def\endroster{\relaxnext@
 \revert@envir\endroster % restore \envir@end
 \par\leftskip@% End the paragraph, and restore the \leftskip.
 \global\rosteritemwd16\p@ % restore default value
 \penalty-50 \vskip-\parskip\smallskip% Add a good break and
 \DN@{\ifx\next\Runinitem\let\next@\relax
  \else\nextRunin@false\let\item\plainitem@% Otherwise, set
   \ifx\next\par% moreover, if \endroster is followed by \par,
    \DN@\par{\everypar\expandafter{\the\everypartoks@}}%
   \else% but if the \endroster isn't followed by a new paragraph,
    \DN@{\noindent\everypar\expandafter{\the\everypartoks@}}%
  \fi\fi\next@}%
 \FN@\next@}
\newcount\rosterhangafter@
\def\Runinitem#1\roster\runinitem{\relaxnext@
  \envir@stack\endroster
 \rostercount@\z@
 \def\item{\FN@\rosteritem@}%
 \def\runinitem@{#1}%
 \DN@{\ifx\next[\let\next\nextii@\else\ifx\next"\let\next\nextiii@
  \else\let\next\nextiv@\fi\fi\next}%
 \DNii@[##1]{\rostercount@##1\relax
  \def\item@{\therosteritem{\number\rostercount@}}\nextv@}%
 \def\nextiii@"##1"{\def\item@{{\rm##1}}\nextv@}%
 \def\nextiv@{\advance\rostercount@\@ne
  \def\item@{\therosteritem{\number\rostercount@}}\nextv@}%
 \def\nextv@{\setbox\z@\vbox
  {\ifnextRunin@\noindent\fi
  \runinitem@\unskip\enspace\item@~\par
  \global\rosterhangafter@\prevgraf}%
  \firstitem@false% Set \firstitem@false for future \item's.
  \ifnextRunin@\else\par\fi
  \hangafter\rosterhangafter@\hangindent3\normalparindent
  \ifnextRunin@\noindent\fi
  \runinitem@\unskip\enspace%  Put in all the stored stuff
  \item@~\ifnextRunin@\else\par@\fi% and the \item@, and
  \nextRunin@true\ignorespaces}%% Here's where we set \nextRunin@true.
 \FN@\next@}
\message{footnotes,}
\def\footmarkform@#1{$\m@th^{#1}$}
\let\thefootnotemark\footmarkform@
\def\makefootnote@#1#2{\insert\footins
 {\interlinepenalty\interfootnotelinepenalty
 \eightpoint\splittopskip\ht\strutbox\splitmaxdepth\dp\strutbox
 \floatingpenalty\@MM\leftskip\z@skip\rightskip\z@skip
 \spaceskip\z@skip\xspaceskip\z@skip
 \leavevmode{#1}\footstrut\ignorespaces#2\unskip\lower\dp\strutbox
 \vbox to\dp\strutbox{}}}
\newcount\footmarkcount@
\footmarkcount@\z@
\def\footnotemark{\let\@sf\empty@\relaxnext@
 \ifhmode\edef\@sf{\spacefactor\the\spacefactor}\/\fi
 \DN@{\ifx[\next\let\next@\nextii@\else
  \ifx"\next\let\next@\nextiii@\else
  \let\next@\nextiv@\fi\fi\next@}%
 \DNii@[##1]{\footmarkform@{##1}\@sf}%
 \def\nextiii@"##1"{{##1}\@sf}%
 \def\nextiv@{\iffirstchoice@\global\advance\footmarkcount@\@ne\fi
  \footmarkform@{\number\footmarkcount@}\@sf}%
 \FN@\next@}
\def\footnotetext{\relaxnext@
 \DN@{\ifx[\next\let\next@\nextii@\else
  \ifx"\next\let\next@\nextiii@\else
  \let\next@\nextiv@\fi\fi\next@}%
 \DNii@[##1]##2{\makefootnote@{\footmarkform@{##1}}{##2}}%
 \def\nextiii@"##1"##2{\makefootnote@{##1}{##2}}%
 \def\nextiv@##1{\makefootnote@{\footmarkform@%
  {\number\footmarkcount@}}{##1}}%
 \FN@\next@}
\def\footnote{\let\@sf\empty@\relaxnext@
 \ifhmode\edef\@sf{\spacefactor\the\spacefactor}\/\fi
 \DN@{\ifx[\next\let\next@\nextii@\else
  \ifx"\next\let\next@\nextiii@\else
  \let\next@\nextiv@\fi\fi\next@}%
 \DNii@[##1]##2{\footnotemark[##1]\footnotetext[##1]{##2}}%
 \def\nextiii@"##1"##2{\footnotemark"##1"\footnotetext"##1"{##2}}%
 \def\nextiv@##1{\footnotemark\footnotetext{##1}}%
 \FN@\next@}
\def\adjustfootnotemark#1{\advance\footmarkcount@#1\relax}
\def\footnoterule{\kern-3\p@
  \hrule width5pc\kern 2.6\p@}%      the \hrule is .4pt high
\message{figures and captions,}
\def\captionfont@{\smc}
\def\topcaption#1#2\endcaption{%
  {\dimen@\hsize \advance\dimen@-\captionwidth@
   \rm\raggedcenter@ \advance\leftskip.5\dimen@ \rightskip\leftskip
  {\captionfont@#1}%
  \if\notempty{#2}.\enspace\ignorespaces#2\fi
  \endgraf}\nobreak\bigskip}
\def\botcaption#1#2\endcaption{%
  \nobreak\bigskip
  \setboxz@h{\captionfont@#1\if\notempty{#2}.\enspace\rm#2\fi}%
  {\dimen@\hsize \advance\dimen@-\captionwidth@
   \leftskip.5\dimen@ \rightskip\leftskip
   \noindent \ifdim\wdz@>\captionwidth@
   \else\hfil\fi
  {\captionfont@#1}%
  \if\notempty{#2}.\enspace\rm#2\fi\endgraf}}
\def\@ins{\par\begingroup\def\vspace##1{\vskip##1\relax}%
  \def\captionwidth##1{\captionwidth@##1\relax}%
  \setbox\z@\vbox\bgroup} % start a \vbox
\message{miscellaneous,}
\def\block{\RIfMIfI@\nondmatherr@\block\fi
       \else\ifvmode\noindent$$\predisplaysize\hsize
         \else$$\fi
  \def\endblock{\par\egroup$$}\fi
  \vbox\bgroup\advance\hsize-2\indenti\noindent}
\def\endblock{\par\egroup}
\def\cite#1{\rom{[{\citefont@\m@th#1}]}}
\def\citefont@{\rm}
\def\rom#1{\leavevmode
  \edef\prevskip@{\ifdim\lastskip=\z@ \else\hskip\the\lastskip\relax\fi}%
  \unskip
  \edef\prevpenalty@{\ifnum\lastpenalty=\z@ \else
    \penalty\the\lastpenalty\relax\fi}%
  \unpenalty \/\prevpenalty@ \prevskip@ {\rm #1}}
\message{references,}
\def\refsfont@{\eightpoint}
\newdimen\refindentwd
\setboxz@h{\refsfont@ 00.\enspace}
\refindentwd\wdz@
\outer\def\Refs{\add@missing\endroster \add@missing\endproclaim
 \let\savedef@\Refs \let\Refs\relax % because of \outer-ness
 \def\Refs##1{\restoredef@\Refs
   \if\notempty{##1}\penaltyandskip@{-200}\aboveheadskip
     \begingroup \raggedcenter@\headfont@
       \ignorespaces##1\endgraf\endgroup
     \penaltyandskip@\@M\belowheadskip
   \fi
   \begingroup\def\envir@end{\endRefs}\refsfont@\sfcode`\.\@m
   }%
 \nofrillscheck{\csname Refs\expandafter\endcsname
  \frills@{{References}}}}
\def\endRefs{\par % This will check for a missing \endref, also
  \endgroup}
\newif\ifbook@ \newif\ifprocpaper@
\def\nofrills{%
  \expandafter\ifx\envir@end\endref
    \let\do\relax
    \xdef\nofrills@list{\nofrills@list\do\curbox}%
  \else\errmessage{\Invalid@@ \string\nofrills}%
  \fi}%
\def\defaultreftexts{\gdef\edtext{ed.}\gdef\pagestext{pp.}%
  \gdef\voltext{vol.}\gdef\issuetext{no.}}
\defaultreftexts
\def\ref{\par
  \begingroup \def\envir@end{\endref}%
  \noindent\hangindent\refindentwd
  \def\par{\add@missing\endref}%
  \global\let\nofrills@list\empty@
  \refbreaks
  \procpaper@false \book@false \moreref@false
  \def\curbox{\z@}\setbox\z@\vbox\bgroup
}
\let\keyhook@\empty@
\def\endref{%
  \setbox\tw@\box\thr@@
  \makerefbox?\thr@@{\endgraf\egroup}%
  \endref@
  \endgraf
  \endgroup
  \keyhook@
  \global\let\keyhook@\empty@ % \global to conserve save stack
}
\def\key{\gdef\key{\makerefbox\key\keybox@\empty@}\key} \newbox\keybox@
\def\no{\gdef\no{\makerefbox\no\keybox@\empty@}%
  \gdef\keyhook@{\refstyle C}\no}
\def\by{\makerefbox\by\bybox@\empty@} \newbox\bybox@
\let\manyby\by % for backward compatibility
\def\bysame{\by\hbox to3em{\hrulefill}\thinspace\kern\z@}
\def\paper{\makerefbox\paper\paperbox@\it} \newbox\paperbox@
\def\paperinfo{\makerefbox\paperinfo\paperinfobox@\empty@}%
  \newbox\paperinfobox@
\def\jour{\makerefbox\jour\jourbox@
  {\aftergroup\book@false \aftergroup\procpaper@false}} \newbox\jourbox@
\def\issue{\makerefbox\issue\issuebox@\empty@} \newbox\issuebox@
\def\yr{\makerefbox\yr\yrbox@\empty@} \newbox\yrbox@
\def\pages{\makerefbox\pages\pagesbox@\empty@} \newbox\pagesbox@
\def\page{\gdef\pagestext{p.}\makerefbox\page\pagesbox@\empty@}
\def\ed{\makerefbox\ed\edbox@\empty@} \newbox\edbox@
\def\eds{\gdef\edtext{eds.}\makerefbox\eds\edbox@\empty@}
\def\book{\makerefbox\book\bookbox@
  {\it\aftergroup\book@true \aftergroup\procpaper@false}}
  \newbox\bookbox@
\def\bookinfo{\makerefbox\bookinfo\bookinfobox@\empty@}%
  \newbox\bookinfobox@
\def\publ{\makerefbox\publ\publbox@\empty@} \newbox\publbox@
\def\publaddr{\makerefbox\publaddr\publaddrbox@\empty@}%
  \newbox\publaddrbox@
\def\inbook{\makerefbox\inbook\bookbox@
  {\aftergroup\procpaper@true \aftergroup\book@false}}
\def\procinfo{\makerefbox\procinfo\procinfobox@\empty@}%
  \newbox\procinfobox@
\def\finalinfo{\makerefbox\finalinfo\finalinfobox@\empty@}%
  \newbox\finalinfobox@
\def\miscnote{\makerefbox\miscnote\miscnotebox@\empty@}%
  \newbox\miscnotebox@
\def\toappear{\miscnote to appear}
\def\lang{\makerefbox\lang\langbox@\empty@} \newbox\langbox@
\newbox\morerefbox@
\def\vol{\makerefbox\vol\volbox@{\ifbook@ \else
  \ifprocpaper@\else\bf\fi\fi}}
\newbox\volbox@
\newbox\holdoverbox
\def\makerefbox#1#2#3{\endgraf
  \setbox\z@\lastbox
  \global\setbox\@ne\hbox{\unhbox\holdoverbox
    \ifvoid\z@\else\unhbox\z@\unskip\unskip\unpenalty\fi}%
  \egroup
  \setbox\curbox\box\ifdim\wd\@ne>\z@ \@ne \else\voidb@x\fi
  \ifvoid#2\else\Err@{Redundant \string#1; duplicate use, or
     mutually exclusive information already given}\fi
  \def\curbox{#2}\setbox\curbox\vbox\bgroup \hsize\maxdimen \noindent
  #3}
\def\refbreaks{%
  \def\refconcat##1{\setbox\z@\lastbox \setbox\holdoverbox\hbox{%
       \unhbox\holdoverbox \unhbox\z@\unskip\unskip\unpenalty##1}}%
  \def\holdover##1{%
    \RIfM@
      \penalty-\@M\null
      \hfil$\clubpenalty\z@\widowpenalty\z@\interlinepenalty\z@
      \offinterlineskip\endgraf
      \setbox\z@\lastbox\unskip \unpenalty
      \refconcat{##1}%
      \noindent
      $\hfil\penalty-\@M
    \else
      \endgraf\refconcat{##1}\noindent
    \fi}%
  \def\break{\holdover{\penalty-\@M}}%
  \let\vadjust@\vadjust
  \def\vadjust##1{\holdover{\vadjust@{##1}}}%
  \def\newpage{\vadjust{\vfill\break}}%
}
\def\refstyle#1{\uppercase{%
  \if#1A\relax \def\keyformat##1{[##1]\enspace\hfil}%
  \else\if#1B\relax
    \def\keyformat##1{\aftergroup\kern
              \aftergroup-\aftergroup\refindentwd}%
    \refindentwd\parindent
 \else\if#1C\relax
   \def\keyformat##1{\hfil##1.\enspace}%
 \fi\fi\fi}% end of \uppercase
}
\refstyle{A}
\def\finalpunct{\ifnum\lastkern=\m@ne\unkern\else.\fi
       \refquotes@\refbreak@}%
\def\continuepunct#1#2#3#4{}%
\def\endref@{%
  \keyhook@
  \def\nofrillscheck##1{%
    \def\do####1{\ifx##1####1\let\frills@\eat@\fi}%
    \let\frills@\identity@ \nofrills@list}%
  \ifvoid\bybox@
    \ifvoid\edbox@
    \else\setbox\bybox@\hbox{\unhbox\edbox@\breakcheck
      \nofrillscheck\edbox@\frills@{\space(\edtext)}\refbreak@}\fi
  \fi
  \ifvoid\keybox@\else\hbox to\refindentwd{%
       \keyformat{\unhbox\keybox@}}\fi
  \ifmoreref@
    \commaunbox@\morerefbox@
  \else
    \kern-\tw@ sp\kern\m@ne sp
  \fi
  \ppunbox@\empty@\empty@\bybox@\empty@
  \ifbook@ % Case 1: \book etc.
    \commaunbox@\bookbox@ \commaunbox@\bookinfobox@
    \ppunbox@\empty@{ (}\procinfobox@)%
    \ppunbox@,{ vol.~}\volbox@\empty@
    \ppunbox@\empty@{ (}\edbox@{, \edtext)}%
    \commaunbox@\publbox@ \commaunbox@\publaddrbox@
    \commaunbox@\yrbox@
    \ppunbox@,{ \pagestext~}\pagesbox@\empty@
  \else
    \commaunbox@\paperbox@ \commaunbox@\paperinfobox@
    \ifprocpaper@ % Case 2: \paper ... \inbook
      \commaunbox@\bookbox@
      \ppunbox@\empty@{ (}\procinfobox@)%
      \ppunbox@\empty@{ (}\edbox@{, \edtext)}%
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\def\rightheadline{\eightpoint\kern3.5em
IN DIMENSION 2, COMPLETE IDEALS ARE MULTIPLIER IDEALS\kern3.1em\the\pageno}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\def\OX{{\O_{\mkern-2.5mu X}}}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\topmatter


\subjclass  13B22, 13H05 \endsubjclass

\title {Integrally closed finite-colength ideals\\ in two-dimensional 
regular local rings\\ are multiplier ideals}
\endtitle

\author Joseph Lipman and Kei-ichi Watanabe\endauthor


%%%% JOE'S DATA %%%%


\address{Dept\. of Mathematics, Purdue University,
 W. Lafayette IN 47907, USA} \endaddress

\email{lipman\@math.purdue.edu}\endemail



%%%% KEIICHI'S DATA %%%% 

\address{Dept\. of Mathematics, Nihon University, Sakura Josui 3-25-40,
Setagaya, Tokyo 156-8550, Japan  } \endaddress

\email{watanabe\@math.chs.nihon-u.ac.jp}   \endemail



%%%%% THANKS %%%%%

\thanks First author partially supported by the National Security
Agency. Second author partially supported by Grants-in-Aid
in Scientific Researches, 13440015,  13874006; and his stay at MSRI was
supported by the Bunri Fund, Nihon University.
Both authors are grateful to MSRI for providing the environment
without which this work would not have begun.
Research at MSRI is supported in part by NSF grant DMS-9810361.
\vadjust{\kern1pt} 
\endthanks

\abstract{
Multiplier ideals in commutative rings are certain integrally closed ideals 
with properties that lend themselves to highly interesting applications. 
But how special are they among integrally closed ideals in general? 
In this note we show that in a two-dimensional regular local ring with 
algebraically closed residue field, in the class of ideals containing a 
power of the maximal ideal there is in fact no
difference between "multiplier" and "integrally closed" (or "complete.")
However, among "integral" multiplier ideals (or "adjoint ideals") the only
simple complete ideals are those of order one.}
\endabstract

\endtopmatter

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\document

\subheading{Introduction} There has arisen in recent years a substantial
body of work on ``multiplier ideals'' in commutative rings (see \cite{La}). 
Multiplier ideals are integrally closed ideals with properties that lend
themselves to highly interesting applications. One is tempted then to
ask just how special multiplier ideals are
among integrally closed ideals in general. 

In this note we show that in a two-dimensional regular local ring $R$
with maximal ideal $\frak m$ such that the residue field $R/\frak m$
is algebraically closed%
\footnote{This last condition can most likely be dropped, but we don't
want to get involved with the resulting technicalities.} 
there is in fact no difference between
multiplier ideals and integrally closed ideals, at least when we deal
with finite-colength ideals (i.e., those containing a power of $\frak m$):
\looseness=-1

\medskip
\noindent
{\bf Main \kern-1pt Result.\kern.5pt}~
{\it Every integrally closed finite-colength\/ $R$-ideal is a multiplier~
ideal.}\looseness=-1

\medskip
We do not know at present whether such a statement holds true for 
arbitrary integrally closed $R$-ideals, let alone  higher
dimensions.

\smallskip
Throughout, $(R,\frak m)$ will be as above. For convenience we say
``complete~ideal" instead of 
``integrally closed finite-colength $R$-ideal."
 

\subheading{1. Geometric formulation of the problem} The goal of this
section is to develop the geometric criterion Corollary 1.4.2
for an ideal to be a multiplier ideal, while laying the groundwork
for the proof in the next section that every complete ideal 
satisfies that criterion.

We begin by recalling
some preliminary definitions and known results. (For some historical
pointers to the development of the theory of complete ideals see the second
paragraph on the first page of \cite{L3}.)


For any complete
ideal $I$ there exists a {\it log resolution,\/} i.e.,  a proper birational
map $f:X\to \Spec(R)$ with $X$ a regular scheme such that the
$\OX$-ideal~$I\OX$ is {\it invertible\/.} A quick way to see this,
with $f$ a composition of maps obtained by blowing up closed points,
is via the Hoskin-Deligne formula \cite{L2, p.\,222, Thm.\, 3.1}, which shows  
that the length of the ``transform'' of~$I\/$ decreases strictly 
with each blowup---until it vanishes, at which point $I$ generates an 
invertible\- ideal sheaf. 

We denote by $E^1,E^2,\dots,E^s$ the integral (i.e.,  reduced 
and irreducible) components of the
closed fiber $f^{-1}\{\frak m\}$. They are isomorphic, as schemes, to the
projective line $\Bbb P^1_{\!R/\frak m}$, and any two of them intersect
transversally. (This can easily be shown by induction on the number of
blowups making up $f\<$.) Each $E^i$ gives rise to the discrete valuation
$v_{\<\<E^i}^{}$ whose valuation ring is the local ring on~$X$ of the
generic point of~$E^i$.

We define the group ${\Dive(X)}$ of (exceptional) {\it
$f$-\kern.7pt divisors\/} to be the free
abelian group on the set $\{E^1,\dots,E^s\}$. The invertible sheaf 
$\OX(E)$ associated to an $f$-\kern.7pt divisor~$E$ is defined in the standard way.
We will use repeatedly, without explicit mention, the fact that
$$
\Gamma\bigl(X\<,\>\OX(a_1E^1+\cdots a_sE^s)\bigr)=\{\,x\in R \mid 
v^{}_{\<E^i}\<(x)\ge -a_i \text{ for  } 1\le i\le s\,\},
$$
an $R$-ideal containing some power of $\frak m$.


An $f$-\kern.7pt divisor $E=a_1E^1+\cdots a_sE^s$ is {\it effective\/} if all 
the integers $a_i$ \hbox{are $\ge 0$}, or equivalently, if $\OX(-E)\subset\OX$. Such an~$E\/$ can be regarded as a
one-dimensional subscheme of $X\<$, projective over $R/\frak m$, 
with structure sheaf $\OE$ fitting in a natural exact sequence\looseness=-1
$$
0\to \OX(-E)\to \OX\to \OE\to 0.
$$


For any invertible $\OX$-module $\L$ the {\it intersection
number\/} $\L\cdot E$ of $\L$ with an effective $E$ is the degree of the
invertible $\OE$-module $\L_E\set \L\otimes\OE$, i.e.,
$$
\L\cdot E =\chi_{\<E}^{} \O_{\mkern-2.5mu E}^{} - \chi_{\<E}^{}\> \L_E^{-1}
$$
where $\chi_{\<E}^{}\>\Cal M$ denotes the Euler characteristic of a coherent
$\OE$-module~$\Cal M$
($E$~being viewed as a projective curve over $R/\frak m$.)

For any $f$-\kern.7pt divisor $F$ set 
\hbox{$F\<\<\cdot\<\< E\>\>\set \OX(F)\<\<\cdot\<\< E$}. This
version of intersection number extends uniquely to a  $\Bbb Z$-valued 
symmetric bilinear form on ${\Dive(X)}$ (see e.g., \cite{L1, \S13}).

An $f$-\kern.7pt divisor $F$  is said to be numerically effective, 
{\it nef\/} for short,
if \hbox{$F\cdot E^i\ge 0$} for all $i=1,2,\dots,s\,$ 
$(\Rightarrow  F\cdot E\ge 0$ for all effective $E\in\Dive^+(X)$). $F$ is said
to be {\it antinef\/} if $-F$ is nef. 

The following basic result is contained in \cite{L1, p.\,220,
Thm.\,(12.1)}. 

\proclaim{Theorem 1.1} 
An\/ $f$-\kern.7pt divisor\/ $F$ is nef if and only if\/ $\O(F)$ is generated 
by its global sections.
\endproclaim


\proclaim{Corollary 1.1.1} 
If\/ $F$ is antinef then\/ $F$ is effective.
\endproclaim

That's because
$\OX(-F)=\Gamma\bigl(X\<,\OX(-F)\bigr)\OX\subset\OX$. (A simpler proof,
not using Theorem 1.1, can be found in \cite{L1, p.\,238}.) 

\smallskip\goodbreak
If $I$ is complete and $I\OX$ is invertible then 
$I=\Gamma(X,I\OX)$, whence:
\proclaim{Corollary 1.1.2} {\rm(Cf.~\cite{L1, \S18}.)}
Sending\/ $E$   to\/ $\Gamma\bigl(X\<, \OX(-E)\bigr)$ is an isomorphism 
from the $($additive\/$)$ monoid of antinef\/ $f$-\kern.7pt divisors
to the 
$($multiplicative\/$)$
monoid of those complete\/ 
$R$-ideals\/~$I$ such that\/ $I\OX$ is invertible.\footnotemark  
\endproclaim
\footnotetext{It is a theorem of Zariski that a product of two complete
ideals is still complete \cite{ZS, p.\,385, Thm. 2$'$},
\cite{L1, p.\,209, Thm.\,(7.1)}.}



It is simple to show, by induction on the number of blowups making up~
$f$, that the intersection matrix $(E^i\cdot E^j)$ has determinant
$\pm 1$. Hence for each~$i$ there is a unique $f$-\kern.7pt divisor $G_i$ such
that $G_i\cdot E^j=0$ unless $j=i$, in which case $G_i\cdot
E^j=-1$; and for any $f$-\kern.7pt divisor $E$ it holds that 
$\>-E=\sum_{i=1}^s (E.E^i)G_i$.
Thus the monoid of antinef divisors is freely generated by
these~$G_i$; in other words, ``unique factorization'' holds in this
monoid---and therefore in the monoid of complete ideals to which, by
Corollary 1.1.2, it is isomorphic.

A complete ideal $P$ is {\it simple\/} if whenever $P=IJ\>$ then either $I$ or
$J$ is the unit ideal.

\proclaim{Corollary 1.1.3} {\rm (Zariski, \cite{ZS, p.\,386, Thm.\,3}.)} Every complete ideal is, in a unique way,
the product of simple complete ideals.
\endproclaim

(For deducing this corollary it helps to
note that for any product $IJ$ of 
ideals, if $IJ\OX$ is invertible then $I\OX$ and $J\OX$ are both invertible.)

\proclaim{Corollary 1.1.4} If\/ $I$\vadjust{\kern1pt} is a complete ideal such that\/ 
$I\OX=\OX(-E)$
is invertible and if\/ 
$P_i\set\Gamma\bigl(X\<,\OX(-G_i)\bigr)$\vadjust{\kern.7pt} is the simple complete
ideal corresponding to the above\/ $G_i,$ then\/ $I=\prod_{i=1}^s P_i^{E.E^i}\<\<.$
In particular, $P_i$ divides\/ $I\iff E\cdot E^i\ne 0$.
\endproclaim

Moreover, the valuations $v^{}_{\<\<E^i}$ associated to those $E^i$ such that 
$E\cdot E^i\ne 0$ are precisely the Rees valuations of $I\>$ (i.e.,
those valuations whose valuation ring is the local ring of the
generic point of some
integral component of the closed fiber of the normalized blowup of
$I$). (See \cite{L4, p.\, 300, Prop.\,(4.4)}.)

\smallskip
The following Lemma%
\footnote{related to Enriques's ``principle of discharge''
 \cite{Z, p.\,28},}
 will be needed. We write 
$\sum_i a_iE^i\ge \sum_ib_iE^i$ if $a_i\ge b_i$ for all $i=1,2,\dots,s$.

\proclaim{Lemma 1.2}
Let\/ $E=\sum_i a_iE^i$ and let\/ $I$ be the\/ $R$-ideal\/
$\Gamma(X\<,\OX(-E))$. Then\/ $I\OX$ is invertible. Equivalently, 
there exists an antinef\/ $E^-\ge E$
such that\/ $I=\nmb\Gamma(X\<,\OX(-E^-))$.
\endproclaim

Corollary 1.1.2 implies that this $E^-$ must be the least antinef
$f$-\kern.7ptdivisor $\ge E$.

\proof One may assume $X\ne\Spec(R)$. There exists {\it some\/} 
antinef $F\ge E$: pick~$n$ such that $\frak m^n\subset I$,
then define $F$ by $\frak m^n\OX=\OX(-F)$. Among all 
antinef $F=\sum b_iE^i\ge E$ choose one---call 
it $F_E$---for which
$\sum_i (b_i-a_i)$ has minimal value,  denoted $\sigma^{}_{\<\<E}$.

Procede by induction on $\sigma^{}_{\<\<E}$.
Suppose $\sigma^{}_{\<\<E}>0$ (otherwise there is nothing to prove), and that
the Lemma holds for all $E'$ with $\sigma^{}_{\<\<E'}<\sigma^{}_{\<\<E}$.
With $F\set F_E$  as above, there is an $i$ such
that $E\cdot E^i>0\ge F\cdot E^i$, and since 
$E^j\cdot E^i\ge0$ when $j\ne i$ (clearly) and
 $E^i\cdot E^i<0$,
\footnote{It is well-known, going back to Du Val, that the intersection matrix
$(E^i\cdot E^j)$ is negative definite, see e.g.,
\cite{L1, p.\,224, Lemma (14.1)}.}
 therefore $b_i>a_i\>$, whence
$F\ge E+E^i$. So $\sigma^{}_{\<\<E+E^i}<\sigma^{}_{\<\<E}\>$, and therefore
$\Gamma(X\<,\>\OX(-E-E^i))\OX$ is invertible. It suffices then to verify that
$
\Gamma\bigl(X\<,\>\OX(-E-E^i)\bigr)=I,
$
by applying the left-exact functor $\Gamma(X\<,-)$ to the 
natural exact sequence
$$
0\longrightarrow \OX(-E-E^i)\longrightarrow \OX(-E) 
\longrightarrow \OX(-E)\otimes \O_{\<\<E^i}
$$
and observing that since  $\OX(-E)\otimes\O_{\<\<E^i}$ 
has degree $-E\cdot E^i<0$, therefore
$$
\Gamma\bigl(X\<,\>\OX(-E)\otimes \O_{\<\<E^i}\<\bigr)=
\Gamma\bigl(E^i\<,\>\OX(-E)\otimes \O_{\<\<E^i}\<\bigr)=0.
\hbox to 0pt{$\hskip 40pt\square$\hss}
$$ 
\enddemo 

\nextpart{1.3} (Canonical divisors.) Let 
$Y\overset{g\,}\to{\to} X\overset{f\,}\to{\to}\Spec(R)$
be proper birational maps with $Y$ and $X$ regular schemes. 
By a theorem of Zariski and Abhyankar (see, e.g., \cite{L1, p.\,204,
Thm.\,(4.1)}) both $f$ and~$g$ are compositions of point blowups. 


Let $F^1,F^2,\dots,F^t$  be the integral components of 
$(fg)^{-1}\{\frak m\}$.
For an $f$-\kern.7pt divisor~$E$, $g^*\<E$ denotes the $fg$-divisor 
with $F^i$-coefficient $v_{F^i}(\OX(-E)_{x_i})$, where $x_i\in X$ is the $g$-image 
of the generic point of~$F^i$.

Since, as before, the intersection matrix $(F^i\cdot F^j)$ has
determinant~$\pm 1$, there is a unique $fg$-divisor $K_g$ such that
$$
K_g\cdot F^i = 
\cases 
-F^i\<\<\cdot\<\< F^i\> -2 &\quad\text{if $g(F^i)$ is a point}\\
0 &\quad\text{otherwise.}
\endcases
$$
This $K_g$ is called the  {\it canonical divisor
of~$g$}.% 
\footnote{One has $\O(K_{\<g})=H^0(g^!\OX)$ with $g^!$  as
in Grothendieck duality theory \cite{LS, p.\,206, (2.3)}.}
\vadjust{\kern1.5pt}%


\nextpart{1.3.1} \kern-3pt The following easily-checked properties characterize $K_{\<g}$ for
all~$g\>$:\vadjust{\kern1.5pt}\looseness=-1

$\bullet$ If $g$ is the blowup of a closed point $x\in X$ 
then $K_{\<g}=g^{-1}\{x\}$.

$\bullet$ If $Z\overset{h\,\>\>}\to{\to} Y\overset{g\,}\to{\to}
X\overset{f\,}\to{\to}\Spec(R)$ are proper birational maps with 
$Z$, $Y$ and~$X$ regular schemes, then
$$
K_{\<gh}= h^*\<K_{\<g} + K_{\<h}.
$$

\nextpart{1.4} (Multiplier ideals.)
For $E=\sum_i a_iE^i\in\Dive(X)\otimes_{\Bbb Z}\Bbb R\>$ set 
$$
[E\>]=\sum_i \,[a_i]E^i\in\Dive(X)
$$
where $[a_i]$ is the greatest integer $\le a_i$.

\proclaim{Definition 1.4.1}
\rm{Let $I$ be a complete $R$-ideal,  $h:Y\to \Spec(R)$ a log
resolution of\/~$I$,
say $I\O_Y=\O_Y(-G)$, and let $c$ be a positive real number. 
The {\it multiplier ideal\/} $\J(cI)$ is defined to be}
$$
\J(cI)\set \Gamma\bigl(Y, \O(K_{\<h}-[c\>G\>])\bigr).
$$
{\rm Thus, by Lemma 1.2, }
$$
\J(cI)\O_Y=\O_Y\bigl(-D\bigr)
$$
\rm{where $D\set\bigl([c\>G\>]-K_{\<h}\bigr)^-$ is the least 
antinef $h$-divisor $\ge[c\>G\>]-K_{\<h}$.}
\endproclaim

For a point blowup $h_1:Y_1\to Y$ one finds via (1.3.1) 
that the log resolution~$h$ can be replaced by the log resolution
$h\smcirc h_1$ without affecting $\J(cI)$. Since any two log
resolutions are dominated by a third, obtained from each of the two by
a sequence of blowups,%
\footnote{By the above-mentioned theorem of Zariski and
Abhyankar, it suffices to principalize some ideal sheaf on one of the
log resolutions by a sequence of point blowups
(``elimination of indeterminacies''), which can be done e.g., via
the Hoskin-Deligne formula, as before.}
it follows that $\J(cI)$ does not depend on the
choice of the log resolution $h$.


\proclaim{Corollary 1.4.2} A complete\/ $R$-ideal\/ $J$ satisfies\/
$J=\Cal J(cI)$ for some\/ $c,I$ iff for some log resolution\/ 
$h:Y\to\Spec(R)$ of\/ $J,$ say\/ $J\O_Y=\O_Y(-F),$ there is an 
antinef\/ $h$-divisor\/~$G$ and a real\/ $c>0$ such that 
$$
F=\bigl([c\>G\>]-K_{\<h}\bigr)^-\,.
\tag 1.4.2.1
$$
\endproclaim

\subheading{2. Proof of Main Result} Let $J$ be a complete $R$-ideal.
We will describe a log resolution $h:Y\to\Spec(R)$ of\/ $J\<$, 
and a $G$ and $c$ as in Corollary 1.4.2, such that if
$J\O_Y=\O_Y(-F)$ then (1.4.2.1) holds. (The number of suitable $(h, G,
c)$ will be enormous.)

Factor $J$ as
$J=\prod_{\ell=1}^{u} P_\ell^{e^{}_{\<\ell}}\ (P_\ell\text{ simple
complete, } e^{}_{\<\ell}>0)$---see Corollary 1.1.3 and the two paragraphs
preceding it.
Let $f:X\to\Spec(R)$ be any log resolution of $J$,
$J\OX=\OX(-F^{\>0})$. We construct a proper birational map $g^{}_{\<N}:Y_N\to X$,
with $Y_N$ regular, for each
$u$-tuple $N\set(n_1,n_2,\dots,n_u)$ of non-negative integers, as follows.

For convenience of expression, we say ``blow up a closed point $x\in X$
generically, $n$ times'' to mean ``blow up $x_0\set x$ to get
$g_1:X_1\to X$, then blow up 
a closed point $x_1$ on $g_1^{-1}x_0$ but not on any other
integral component of the closed fiber of $X_1\to \Spec(R)$ to get
$g_2:X_2\to X_1$, then blow up a closed point $x_2$ on $g_2^{-1}x_1$ but not
on \dots then blow up 
a closed point $x_{n-1}$ on $g_{n-1}^{-1}x_{n-2}$ but not on any other
integral component of the closed fiber of $X_{n-1}\to \Spec(R)$ to get
$g_n:X_n\to X_{n-1}$.''

Then with $g\set g_1\smcirc g_2\smcirc\smcirc\cdots\smcirc g_n$ it
holds that:\vadjust{\kern1.5pt}

\noindent{\bf (2.1)} $g^{-1}x $ is a chain of $n$ integral
curves $D_{1},D_{2},\dots,D_{n}$ such that for $0<i<n$, 
$D_{i}\cdot D_{i+1}=1$ and  $D_{i}\cdot D_{i}=-2$, 
while $D_n\cdot D_n=-1$; and if $|j-i|>1$ then $D_{i}\cdot D_{j}=0$.
\vadjust{\kern1pt} 

(For the proof one can use, e.g., \cite{L1, p.\,229, middle, and p.\,
227, $\alpha)$ and $\beta)$}. Here, and subsequently, the reader may find it useful to do some rough sketches.)

As before, there corresponds to each $D_i$ a simple complete ideal
$P_i\>$; and, we claim, {\it these $P_i$ form a strictly decreasing
sequence\/} $P_1> P_2>\dots>P_n$, with $P_1$ strictly contained in each of
the simple ideals corresponding to the (one or two) integral
components $E^j$ of~$f^{-1}\{\frak m\}$ passing through~$x$. 

Indeed,
let $G_{\<\<j}$ be the \hbox{$f$-\kern.7pt divisor} such that $G_{\<\<j}\cdot E^j=-1$
and $G_{\<\<j}\cdot E=0$ for every other integral component~$E$ of~
$f^{-1}\{\frak m\}$, and let $Q_j$ be the corresponding simple
complete ideal.  It follows from, e.g., \cite{L1,
p.\,227, $\alpha)$ and $\beta)$} that $g_1^*G_{\<\<j}$ is antinef; and the
corresponding simple complete ideal is
$$
\Gamma\bigl(X_1\>, \O_{\!X_1}(-g_1^*G_{\<\<j})\bigr)=
\Gamma\bigl(X\<, \OX(-G_{\<\<j})\bigr)=Q_j\>.
$$
Further, with $E':=g_1^{-1}x$, 
let $G'$  be the $fg_1$-\kern.7pt divisor 
such that $G'\cdot E'=-1$ and
$G'\cdot E''=0$ for every other integral component~$E''$ of
$(fg_1)^{-1}\{\frak m\}$. 
Then $g_1^*G_{\<\<j}+E'$ has intersection number 
$-1$ with $E'$ and $\ge 0$ with each~$E''$. So  $G'-g_1^*G_{\<\<j}-E'$ is
antinef, hence effective (Corollary 1.1.1); and consequently $G'>g_1^*G_{\<\<j}$. Thus the
simple complete ideal $\Gamma\bigl(X_1\>, \O_{\!X_1}(-G')\bigr)$ is strictly
contained in $Q_j$.\looseness=-1 

Continuing in this way we establish the claim.
 
\medbreak

 


Now for each simple factor $P_\ell$  of $J\<$, there is a unique
integral component $E^\ell$ of $f^{-1}\{\frak m\}$ such
that $P_\ell\>\OX\cdot E^\ell =1$ and $P_\ell\>\OX\cdot E=0$ for
any other integral component~$E$; and hence $e^{}_{\<\ell}=J\OX\cdot E^\ell$.
For each $\ell=1,2,\dots,u$, pick $e^{}_{\<\ell}$ distinct closed points 
$x^{}_1,\dots,x_{e^{}_{\<\ell}}$ which
lie on $E^\ell$ but on no $E\ne E^\ell$ and blow up all of
these points generically,
$n^{}_\ell$ times. Then $Y_N$ is the resulting surface, and $g^{}_{\<N}$ is the
composition of all the blowups. It is easily seen that $(Y_N, g^{}_{\<N})$
does not depend (up to isomorphism) on the order in which the chosen
points are blown up---though that won't really be used.
\footnote{The initial $\sum_\ell e^{}_\ell$ 
points could be taken to be
the intersection of the
closed fiber on $X$ and a generic curve 
$C$ in the linear system $|-F|$ (i.e.,
a divisor---having no component in the closed fiber---of 
the form $(j)-F$ with $j$ a generic element of $J$). Then at each stage
the points to be blown up could be taken to be closed points on the
inverse image of~$C$.}


To simplify notation, fix $N$ and set $(Y,g)\set (Y_N, g^{}_{\<N})$ 
and $F\set g^*F^{\>0}$, so that $J\O_Y=\O_Y(-F)$. Also, 
set $h\set fg:Y\to \Spec(R)$.

For an $f$-\kern.7pt divisor $E$, we denote by $E^*$ the proper
transform of $E$ on $Y\<$, i.e., if $g^*E=\sum_i a_iE'_i+\sum_j b_jE''_j$
where each $gE'_i$ is a curve on~$X$ while each $gE''_j$ is a closed point, then
$E^*\set\sum_i a_iE'_i$. 

For each $\ell=1,2,\dots,u$ and 
$x_{\!j^{}_{\<\ell}}\in E^\ell\ (j^{}_{\<\ell}=1,2,\dots, e^{}_\ell)$ let 
$$
\{\,E^\ell_{j^{}_{\<\ell}k^{}_{\<\ell}}
\mid k^{}_{\<\ell}=1,2,\dots,n^{}_\ell \,\}
$$
be the family of integral curves on $Y$ 
whose $g$-image is $x_{\<j^{}_{\<\ell}}$, the
ordering of these curves by the index $k^{}_{\<\ell}$ conforming to
the ordering of the $D$'s in (2.1).

If $a^{}_\ell$ is the $E^\ell$-coefficient of the divisor
$F^{\>0}$, and $b^{}_\ell$ of the divisor $K_{\!f}$, then one finds
(using (1.3.1)) that
$$
\align
F=g^*F^{\>0}&= F^{\>0}{}^* + 
 \sum_\ell \sum_{j^{}_{\<\ell}\<,\>k^{}_{\<\ell}}a^{}_\ell E^\ell_{j^{}_{\<\ell}k^{}_{\<\ell}},\\
g^*K_{\!f}&=K_{\!f}^*+\sum_\ell
\sum_{j^{}_{\<\ell}\<,\>k^{}_{\<\ell}} b^{}_\ell E^\ell_{j^{}_{\<\ell}k^{}_{\<\ell}},\\
K_{\<g}&=\sum_\ell \sum_{j^{}_{\<\ell}\<,\>k^{}_{\<\ell}}  
  k^{}_\ell E^\ell_{j^{}_{\<\ell}k^{}_{\<\ell}}.
\endalign
$$

Set $G\set F+K_{\<g}$. Expanding $F$ and $K_{\<g}$ as above, noting that
$g^*\<D\cdot E^\ell_{j_{\<\ell}k^{}_{\<\ell}}=0$ for any $f$-\kern.7pt divisor~$D$
\cite{L1, p.\,227, $\beta)$}, and 
using (2.1), one calculates that for every integral
component $E$ of $h^{-1}\{\frak m\}$, $G\cdot E=0$ unless $E$ is one
of the curves 
$E^\ell_{j^{}_{\<\ell}n^{}_{\<\ell}}$ at the end of the chains
emanating from the $\sum_\ell e_{\<\ell}$ originally
chosen points (i.e., $\>g(E)$ is a point and $E\cdot E=-1$), in which
case $G\cdot E=-1$. So $G$ is antinef, and by Corollary 1.1.4,
$$
I\set\Gamma\bigl(Y, \O_{Y}(-G)\bigr)
$$
is the product of the simple complete ideals corresponding to these
$\sum_\ell e^{}_{\<\ell}$  curves having self-intersection $-1$.

\smallskip

Here is the key technical point:
\proclaim{Lemma 2.2} For all sufficiently small\/ $\epsilon>0$, there
exists\/ $N$ such that 
$$
(1+\epsilon)G-K_{\<h}=F + A\qquad\bigl([A]\le 0\bigr)\tag 2.2.1
$$
where the coefficient of\/ $[A]$ at each\/ $E^\ell_{j^{}_{\<\ell}n^{}_{\<\ell}}$ \!is\/ $\>0$.
\endproclaim

\proof
Using (1.3.1), one transforms (2.2.1) into the equality
$$
\epsilon(F+K_{\<g})-g^*\<K_{\!f}= A.
$$

More explicitly (see above)
$$
\epsilon\bigl(F^{\>0}{}^*+ \sum_\ell  \sum_{j^{}_{\<\ell}\<,\>k^{}_{\<\ell}}  
\bigl(a^{}_\ell+k^{}_\ell\bigr)E^\ell_{j^{}_{\<\ell}k^{}_{\<\ell}}\bigr)
-\bigl(K_{\!f}^*+\sum_\ell  
  \sum_{j^{}_{\<\ell}\<,\>k^{}_{\<\ell}}b^{}_{\<\ell}E^\ell_{j^{}_{\<\ell}k^{}_{\<\ell}}\bigr)=A.
$$
So to get (2.2.1) we can choose any $\epsilon>0$ such that the
coefficients of $\epsilon F^{\>0}-K_{\!f}$ are all $<1$, and then look for
$n^{}_\ell$ such that $\epsilon(a^{}_\ell+ k^{}_\ell)-b^{}_\ell<1$
for all $\ell$ and $k^{}_\ell\le n^{}_\ell$, while
$\epsilon(a^{}_\ell+n^{}_\ell)-b^{}_\ell\ge 0$. These conditions mean
precisely that $n^{}_\ell$ satisfies the inequalities
$$
 1/\epsilon + b^{}_\ell/\epsilon -a^{}_\ell > n^{}_\ell
  \ge b^{}_\ell/\epsilon -a^{}_\ell
\quad (\ell=1,2,\dots,u).
$$ 
Clearly,  such integers $n^{}_\ell$ can be found if $\epsilon<1$.
\endproof


For  $c=1+\epsilon$ and $N$ satisfying Lemma 2.2, and with
$h:Y\to \Spec(R)$ and $F$, $G$, 
as before, we have
$$
F'\set \bigl([c\>G\>]-K_{\<h}\bigr)^-\le F,
$$
so that 
$$
J'\set \Gamma\bigl(Y\<, \O(-F')\bigr)\supset J=\Gamma\bigl(Y\<,
\O(-F)\bigr).
$$
Let us verify that $J'=J$, thereby proving the main  result.


Recall that the valuations $v^{}_\ell\set v^{}_{E^\ell}$ are
just the Rees valuations of~$J$. (See the remark following Corollary 1.1.4). So 
$$
J=\{\,\rho\in R\mid v^{}_\ell(\rho) \ge v^{}_\ell(J) \text{ for all
}\ell=1,2,\dots,u\,\}.
$$

Thus we need only show that for each $\ell$, the $E^\ell{}^*\<$-coefficient
$a'_\ell$ of $F'$ is the same as that of $F$ (namely $a^{}_\ell$).
Let us say that $\ell$ is ``good'' if $a'_\ell=a^{}_\ell$ and ``bad''
if $a'_\ell<a^{}_\ell$.

If $\ell$  is good then since $F'\le F$ 
therefore $F'\cdot E^\ell{}^*\le F\cdot E^\ell{}^*$. 
Corollary~1.1.4 shows then that $J'$ is divisible by $P_\ell^{e^{}_\ell}$. 

Suppose $\ell$ is bad.
For $j\in[1,e^{}_{\<\ell}\>]$ and with $a'_{jk}$ 
the $E^\ell_{jk}$-coefficient of~$F'$
it is easily seen that 
$$
a'_\ell=:a'_{j 0}\le a'_{j 1}\le a'_{j 2}\le\dots\le a'_{jn^{}_\ell}=a^{}_\ell.
$$
So there is a $k\in[1,n^{}_\ell]$ such that 
$
a'_{j,\>k-1}< a'_{jk}= a'_{j,\> k+1}=\dots=a^{}_\ell.
$
Then 
$$
F'\cdot E^\ell_{kl}=
\cases
a'_{j,\>k-1}-2a'_{jk}+a'_{j,\>k+1}<0\quad&\text{if }k<n^{}_\ell\>,\\
a'_{j,\>n^{}_\ell-1}-a'_{jn^{}_\ell}<0   \quad&\text{if }k=n^{}_\ell\>.
\endcases
$$
From Corollary 1.1.4 and the remarks after 2.1, one deduces that $J'$
is divisible by a simple complete ideal $P'_{\ell j}<P_\ell$. This
being so for all $j$, and the $P'_{\ell j}$ being distinct (Corollary 1.1.2), 
it follows from Corollary 1.1.3 that $J'$ is divisible by $P'_{\ell
1}P'_{\ell 2}\cdots P'_{\ell e^{}_\ell} < P_\ell^{e^{}_\ell}$.

Thus (by Corollary 1.1.3 again) the existence of a bad $\ell\>$ 
leads to a factorization of $J'$ which contradicts $J'\supset J$. So
every $\ell$ is good, and $J'=J$.
$\quad\square$

\remark{Remarks} 1. By the choice of $\epsilon$, the
$E^\ell$-\kern.7pt coefficient of $\epsilon F^{\>0}-K_{\!f} $ is $<1$, i.e.,
$\epsilon a^{}_\ell-b^{}_\ell<1$, i.e., 
$b^{}_\ell/\epsilon - a^{}_\ell>-1/\epsilon$. It could happen that  
$b^{}_\ell/\epsilon - a^{}_\ell< 0$ for all $\ell$. In this case one
can take $N=(0,0,\dots,0)$, and then $J=\J((1+\epsilon)J)$.\vadjust{\kern1pt}


2. The proof shows that if $J$ is a simple complete ideal then there
is a simple complete ideal $P\subset J$ and a $c>0$ such that
$J=\J(cP)$.

3. Since the $c=1+\epsilon$ we have considered can be arbitrarily
close to 1, one might ask if it is possible for $c$ actually to be 1.
(This would be the case studied in \cite{L5}, where $\J(I)$ is called the
{\it adjoint ideal\/} of $I$.) 

\goodbreak
For simple complete $J\<$, the answer is given by:

\proclaim{Proposition 2.3} A simple complete ideal\/ $J$ is of the
form\/~$\J(I)$ for some~\/ $I$ $\iff J\not\subset \frak m^2 \iff 
 J=(a,b^n)R$ for some integer\/ $n>0$ and\/ $a,b\in R$ such that\/ 
$(a,b)R=\frak m$.
\endproclaim

\proof
The last $\iff$ holds because $J\not\subset\frak m^2$ means
that $J$ contains an element~$a$ such that $R/aR$ is a discrete
valuation ring. Moreover, if $(a,b)R=\frak m$ and
$z\in R$ is integral over $J=(a,b^n)R$ then the canonical image of $z$ in
$R/aR$ is integral over---and hence is a multiple of---that of $b^n$,
whence $z\in J\<$, and thus $J$ is complete (and clearly simple). And it
is a simple exercise to show that for such a $J\<$, $J=\J(J^2)$. (One
could use \cite{L5, p.\,749, Prop.\,(3.1.2)}.)\vadjust{\kern1.5pt}

For any simple complete $J\<$, on the minimal log resolution
$f:X\to\Spec(R)$ there is, as we have seen, a unique integral component~$E$ of
$f^{-1}\{\frak m\}$ such that  
$J\OX\cdot E\ne 0$. This $E$  must satisfy $E\cdot E=-1$: 
for, there is at least one integral component $E'$ such that
$E'\cdot E'=-1$, namely the closed fibre $f'{}^{-1}\{x'\}$
for the  blowup $f':X\to X'$ of $x'\in X'$ coming last
in some seqence of blowups composing to~$f$; and by minimality of $f$,
$J'\set J\O_{\!X'\!,\>\>x'}$ cannot be invertible, from which one sees, with
$\frak m'$ the maximal ideal of $R'\set\O_{\!X'\<\!,\>\>x'},$ that
 $J'=d\>\frak m'{^t}$ for some $d\in R'$ and $t>0$, whence
$J\OX\cdot E'=t$. (Of course $t=1$.) 

We claim that if $F\cdot F =-2$ for all integral components $F\ne E$
of $f^{-1}\{\frak m\}$, then $J\not\subset\frak m^2\<$.\kern1.7pt%
\footnote
{The converse is part of the exercise at the end of the first
paragraph.}
Indeed, this condition on the $F$'s means that among the base points of
$J$ no two are proximate to the same one, and the conclusion follows 
from \cite{L5, p.\, 301, (3)}. 

Assuming now that our simple $J=\J(I)$, let $h:Y\to \Spec(R)$ be the 
minimal log resolution of $I$, say $I\O_Y=\O_Y(-G)$. As above, we find
that for any integral component $F$ of $h^{-1}\{\frak m\}$, $F\cdot
F=-1\Rightarrow G\cdot F<0$. (We may assume that $F$~is the
closed fiber for the last blowup in some sequence of blowups composing to~$g$.)
Since $K_{\<h}\cdot F=-F\cdot F -2=-1$ for every such~ $F$, it
follows that $G-K_{\<h}$ is antinef, and hence $\O_Y(K_{\<h}-G)=JO_Y$. 

Since
$J\O_Y$ is invertible (Lemma 1.2), the minimality of~$f$ implies that
there exists a $g:Y\to X$ such that
$h=fg$. Let
$F^1{}^*, F^2{}^*,\dots, F^n{}^*$ be the proper transforms on~$Y$ of the
integral components $F^1, F^2,\dots, F^n$ of $f^{-1}\{\frak m\}$ 
other than $E$ (see above). 
Then $F^i{}^*\<\<\cdot F^i{}^*\le F^i\cdot F^i\le-2$
and so $K_{\<h}\cdot F^i{}^*\ge 0$.
But (by \cite{L1, p.\,227, $\beta)$})
$$
K_{\<h}\cdot F^i{}^*\le (K_{\<h}-G)\cdot F^i{}^*=JO_Y\cdot F^i{}^*=JO_X\cdot F^i=0,
$$
and thus $K_{\<h}\cdot F^i{}^*=0$, i.e., $F^i{}^*\<\<\cdot F^i{}^*=-2$, whence, finally,
$F^i\cdot F^i =-2$. The above claim shows then that $J\not\subset\frak
m^2$.
\endproof 
\endremark

\vfill\newpage
\Refs

\widestnumber\key{LS}

\ref\key La \by
R\. Lazarsfeld
\book Positivity in Algebraic Geometry
\bookinfo 
Draft available at \hbox to 65pt{}
{\tt <\;http://www.math.lsa.umich.edu/\~{}rlaz/\;>}
\endref

\ref\key L1 \by
J\. Lipman
\paper Rational singularities, with applications to algebraic surfaces and
unique factorization
\jour Publ\. Math\. IHES \vol 36 \yr 1969
\pages 195--279
\endref

\ref\key L2\bysame
\paper On complete ideals in regular local rings
\inbook Algebraic Geometry and Commutative Algebra, {\rm vol.~I}
\bookinfo in honor of Masayoshi Nagata
\publ Kinokuniya \publaddr Tokyo
\yr  1988 \pages 203--231
\endref

\ref\key L3 \bysame
\paper Adjoints and polars of simple complete ideals in two-dimensional
regular local rings
\jour Bull\. Soc\. Math\. Belgique \vol 45 \yr 1993
\pages 224--244
\endref

\ref\key L4 \bysame
\paper Proximity inequalities for  complete ideals in two-dimensional
regular local rings
\jour Contemporary Mathematics \vol 159 \yr 1994
\pages 293--306
\endref

\ref\key L5 \bysame
\paper Adjoints of ideals in regular local rings
\jour Math.~Research Letters \vol 1 \yr 1994
\pages 739--755
\endref

\ref\key LS \bysame
\ and A\. Sathaye
\paper Jacobian ideals and a theorem of Brian\c con-Skoda
\jour Michigan Math.~J\. \vol 28 \yr 1981
\pages 199--222
\endref

\ref\key Z \by
O\. Zariski
\book Algebraic Surfaces 
\bookinfo (2nd supplemented edition)
\publ Springer-Verlag \publaddr New York \yr 1971
\endref

\ref\key ZS \bysame
 \ and P\. Samuel
\book Commutative Algebra \bookinfo vol.\,2
\publ van Nostrand \publaddr Princeton \yr 1960
\endref

\endRefs

\enddocument