%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Syzygies of Unimodular Lawrence Ideals % % Dave Bayer, Sorin Popescu and Bernd Sturmfels % Plain TeX, uses epsf.tex % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \ifx\begin\undefined\else\global\advance\srcdepth by 1\expandafter\endinput\fi \def\begin{} \newcount\srcdepth \srcdepth=1 \outer\def\bye{\global\advance\srcdepth by -1 \ifnum\srcdepth=0 \def\endcmd{\vfill\supereject\nopagenumbers\par\vfill\supereject\end} \else\def\endcmd{}\fi \endcmd } %=========================================================================% % initialize TeX %=========================================================================% % #1 magnification % #2 text width % #3 text height % #4 page width % #5 page height % #6 vertical adjustment after centering \def\initialize#1#2#3#4#5#6{ \ifnum\srcdepth=1 \magnification=#1 \hsize = #2 \vsize = #3 \hoffset=#4 \advance\hoffset by -\hsize \divide\hoffset by 2 \advance\hoffset by -1truein \voffset=#5 \advance\voffset by -\vsize \divide\voffset by 2 \advance\voffset by -1truein \advance\voffset by #6 \baselineskip=13pt \emergencystretch = 0.05\hsize \fi } \def\print{\initialize{1095} {5.5truein}{8.5truein}{8.5truein}{11truein}{-.0625truein}} \def\powerbook{\initialize{1493} {7.5truein}{10truein}{8.5truein}{11truein}{-.0625truein}} \newif\ifblackboardbold % comment out the following line if AMS msbm fonts aren't available \blackboardboldtrue %=========================================================================% % select fonts %=========================================================================% \font\titlefont=cmbx12 scaled\magstephalf \font\sectionfont=cmbx12 \font\netfont=cmtt9 % allow use of \it and \sl in math subscripts \font\scriptit=cmti10 at 7pt \font\scriptsl=cmsl10 at 7pt \scriptfont\itfam=\scriptit \scriptfont\slfam=\scriptsl % Establish AMS blackboard bold fonts without using amssym.def, amssym.tex \newfam\bboldfam \ifblackboardbold \font\tenbbold=msbm10 \font\sevenbbold=msbm7 \font\fivebbold=msbm5 \textfont\bboldfam=\tenbbold \scriptfont\bboldfam=\sevenbbold \scriptscriptfont\bboldfam=\fivebbold \def\bbold{\fam\bboldfam\tenbbold} \else \def\bbold{\bf} \fi % Establish AMS msam symbols without using amssym.def, amssym.tex \newfam\msamfam \font\tenmsam=msam10 \font\sevenmsam=msam7 \font\fivemsam=msam5 \textfont\msamfam=\tenmsam \scriptfont\msamfam=\sevenmsam \scriptscriptfont\msamfam=\fivemsam \def\msam{\fam\msamfam\tenmsam} \newfam\msbmfam \font\tenmsbm=msam10 \font\sevenmsbm=msam7 \font\fivemsbm=msam5 \textfont\msbmfam=\tenmsbm \scriptfont\msbmfam=\sevenmsbm \scriptscriptfont\msbmfam=\fivemsbm \def\msbm{\fam\msbmfam\tenmsbm} %================================================ % The following macro "\newsymbol " works for MSAM, MSBM und EUFM % automatically using the AMS-Font ID-number 1 for MSAM, 2 for MSBS, % 3 for EUFM similar to the AMS-Macro \newsymbol %------------------------------------------------------------- \newcount\amsfamcount % define new registers to \newcount\classcount % store the digits for conversion \newcount\positioncount \newcount\codecount \newcount\n % store the #2 parameter %------------------------- \def\newsymbol#1#2#3#4#5{ % #1 is the name of the new % % macro to be defined \n="#2 % #2 is the AMS font family number \ifnum\n=1 \amsfamcount=\msamfam\else % translate it to the correct \ifnum\n=2 \amsfamcount=\msbmfam\else % PlainTex font family number \ifnum\n=3 \amsfamcount=\eufmfam \fi\fi\fi \multiply\amsfamcount by "100 % make it the third hex digit \classcount="#3 % #3 is the class number (same as in Plain) \multiply\classcount by "1000 % make it the fourth hex digit \positioncount="#4#5 % #4#5 is the hex position in the fonttable % \codecount=\classcount % add the three numbers to obtain \advance\codecount by \amsfamcount % the Plain TEX identification number \advance\codecount by \positioncount % \mathchardef#1=\codecount} % define #1 to become a character % % with this identification number %=========================================================================% % font size-changing command ("A Beginner's Book of TeX" p35, p275) %=========================================================================% \font\Arm=cmr9 \font\Ai=cmmi9 \font\Asy=cmsy9 \font\Abf=cmbx9 \font\Brm=cmr7 \font\Bi=cmmi7 \font\Bsy=cmsy7 \font\Bbf=cmbx7 \font\Crm=cmr6 \font\Ci=cmmi6 \font\Csy=cmsy6 \font\Cbf=cmbx6 \ifblackboardbold \font\Abbold=msbm10 at 9pt \font\Bbbold=msbm7 \font\Cbbold=msbm5 at 6pt \fi \def\small{% \textfont0=\Arm \scriptfont0=\Brm \scriptscriptfont0=\Crm \textfont1=\Ai \scriptfont1=\Bi \scriptscriptfont1=\Ci \textfont2=\Asy \scriptfont2=\Bsy \scriptscriptfont2=\Csy \textfont\bffam=\Abf \scriptfont\bffam=\Bbf \scriptscriptfont\bffam=\Cbf \def\rm{\fam0\Arm}\def\mit{\fam1}\def\oldstyle{\fam1\Ai}% \def\bf{\fam\bffam\Abf}% \ifblackboardbold \textfont\bboldfam=\Abbold \scriptfont\bboldfam=\Bbbold \scriptscriptfont\bboldfam=\Cbbold \def\bbold{\fam\bboldfam\Abbold}% \fi \rm } %=========================================================================% % single-pass symbolic theorem labeling %=========================================================================% % Because this is a single-pass mechanism with no .aux file, forward % references need to be declared in advance: % \forward{thm:main}{Theorem}{1.1} % This is also the mechanism for "timely" declaration of labels, which % will usually be buried within the corresponding theorem macros. % A warning is issued if a label redeclaration is inconsistent, allowing % forward references to be manually fixed. % \ref{thm:main} produces "Theorem~1.1" % \refs{thm:main} produces "Theorems~1.1" % \refn{thm:main} produces "1.1" % Some TeX adapted from "The Advanced TeXbook" by David Salomon, chapter 9. % Implementers: The code for \forward is subtle. Its second argument must % be provided literally, e.g. "Theorem" rather that "\capitalize{theorem}". % Its third argument must either be literal or a macro that expands % directly to a literal, e.g. "\edef\numtoks{\number\proccount}". % This use of \edef cannot be replaced by \def, which defers expansion. % Failure to follow these rules will cause spurious warnings that forward % references are inconsistent, when they are in fact consistent after % expansion. Note the "Towers of Palo Alto" recreational math problem % involving the iterated use of \expandafter to expand the first argument % to \forwardsub before calling it. \newlinechar=`@ \def\forwardmsg#1#2#3{\immediate\write16{@*!*!*!* forward reference should be: @\noexpand\forward{#1}{#2}{#3}@}} \def\nodefmsg#1{\immediate\write16{@*!*!*!* #1 is an undefined reference@}} \def\forwardsub#1#2{\def\newref{{#2}{#1}}} \def\forward#1#2#3{% \expandafter\expandafter\expandafter\forwardsub\expandafter{#3}{#2} \expandafter\ifx\csname#1\endcsname\relax\else% \expandafter\ifx\csname#1\endcsname\newref\else% \forwardmsg{#1}{#2}{#3}\fi\fi% \expandafter\let\csname#1\endcsname\newref} \def\firstarg#1{\expandafter\argone #1}\def\argone#1#2{#1} \def\secondarg#1{\expandafter\argtwo #1}\def\argtwo#1#2{#2} \def\ref#1{\expandafter\ifx\csname#1\endcsname\relax {\nodefmsg{#1}\bf`#1'}\else \expandafter\firstarg\csname#1\endcsname ~\expandafter\secondarg\csname#1\endcsname\fi} \def\refs#1{\expandafter\ifx\csname#1\endcsname\relax {\nodefmsg{#1}\bf`#1'}\else \expandafter\firstarg\csname #1\endcsname s~\expandafter\secondarg\csname#1\endcsname\fi} \def\refn#1{\expandafter\ifx\csname#1\endcsname\relax {\nodefmsg{#1}\bf`#1'}\else \expandafter\secondarg\csname #1\endcsname\fi} %=========================================================================% % widow control %=========================================================================% % usage: % \widow{.2} % start new page if <.2 page left \def\widow#1{\vskip 0pt plus#1\vsize\goodbreak\vskip 0pt plus-#1\vsize} %=========================================================================% % sections and theorems %=========================================================================% % use \showlabels or \showlabelsabove to display section and theorem labels \def\marginlabel#1{} \def\showlabels{ \font\labelfont=cmss10 at 8pt \def\marginlabel##1{\llap{\labelfont##1\quad}} } \def\showlabelsabove{ \font\labelfont=cmss10 at 6pt \def\marginlabel##1{\rlap{\smash{\raise 10pt\hbox{\labelfont##1}}}} } \newcount\seccount \newcount\proccount \seccount=0 \proccount=0 \def\stdskip{\vskip 9pt plus3pt minus 3pt} \def\stdbreak{\par\removelastskip\penalty-100\stdskip} \def\proof{\stdbreak\noindent{\bf Proof. }} \def\qed{\vrule height 1.2ex width .9ex depth .1ex} \def\Box{ \ifmmode\eqno\qed \else\ifvmode\removelastskip\line{\hfil\qed} \else\unskip\quad\hskip-\hsize \hbox{}\hskip\hsize minus 1em\qed\par \fi\stdbreak\fi} \def\references{ \removelastskip \widow{.05} \vskip 24pt plus 6pt minus 6 pt \leftline{\sectionfont References} \nobreak\stdskip\noindent} \def\ifempty#1#2\endB{\ifx#1\endA} \def\makeref#1#2#3{\ifempty#1\endA\endB\else\forward{#1}{#2}{#3}\fi} \outer\def\section#1 #2\par{ \removelastskip \global\advance\seccount by 1 \global\proccount=0\relax \edef\numtoks{\number\seccount} \makeref{#1}{Section}{\numtoks} \widow{.05} \vskip 24pt plus 6pt minus 6 pt \message{#2} \leftline{\marginlabel{#1}\sectionfont\numtoks\quad #2} \nobreak\stdskip} \def\proclamation#1#2{ \outer\expandafter\def\csname#1\endcsname##1 ##2\par{ \stdbreak \global\advance\proccount by 1 \edef\numtoks{\number\seccount.\number\proccount} \makeref{##1}{#2}{\numtoks} \noindent{\marginlabel{##1}\bf #2 \numtoks\enspace} {\sl##2\par} \stdbreak}} \def\othernumbered#1#2{ \outer\expandafter\def\csname#1\endcsname##1{ \stdbreak \global\advance\proccount by 1 \edef\numtoks{\number\seccount.\number\proccount} \makeref{##1}{#2}{\numtoks} \noindent{\marginlabel{##1}\bf #2 \numtoks\enspace}}} \proclamation{definition}{Definition} \proclamation{lemma}{Lemma} \proclamation{proposition}{Proposition} \proclamation{theorem}{Theorem} \proclamation{corollary}{Corollary} \proclamation{conjecture}{Conjecture} \othernumbered{example}{Example} \othernumbered{remark}{Remark} \othernumbered{construction}{Construction} %========================================================================== % figure, table macros %========================================================================== \def\figure#1{ \global\advance\figcount by 1 \goodbreak \midinsert#1\smallskip \centerline{Figure~\number\figcount} \endinsert} \def\capfigure#1#2{ \global\advance\figcount by 1 \goodbreak \midinsert#1\smallskip \vbox{\small\noindent {\bf Figure~\number\figcount:} #2} \endinsert} \def\capfigurepair#1#2#3#4{ \goodbreak \midinsert #1\smallskip \global\advance\figcount by 1 \vbox{\small\noindent {\bf Figure~\number\figcount:} #2} \vskip 12pt #3\smallskip \global\advance\figcount by 1 \vbox{\small\noindent {\bf Figure~\number\figcount:} #4} \endinsert} % table modified from "A Beginner's Book of TEX" p. 115 \def\baretable#1#2{ \vbox{\offinterlineskip\halign{ \strut\kern #1\hfil##\kern #1 &&\kern #1\hfil##\kern #1\cr #2 }}} \def\gridtablesub#1#2#3{ \vbox{\offinterlineskip\halign{ \strut\vrule\kern #1\hfil##\hfil\kern #2\vrule &&\kern #1\hfil##\kern #2\vrule\cr \noalign{\hrule} #3 \noalign{\hrule} }}} \def\gridtable#1{\gridtablesub{6pt}{3pt}{#1}} \def\gridtablefig#1{\figure{\centerline{\gridtable{#1}}}} \def\capgridtablefig#1#2{\capfigure{\centerline{\gridtable{#1}}}{#2}} \def\capgridtablefigpair#1#2#3#4{\capfigurepair {\centerline{\gridtable{#1}}}{#2}{\centerline{\gridtable{#3}}}{#4}} %=========================================================================% % enable postscript illustrations using epsf.tex %=========================================================================% % Usage: % \draw{70}{fig}{} % draw fig.eps at 70% scale % \draw{999}{fig}{} % draw fig.eps scaled to width of page % Optional third argument can be multiple calls to \figtext; see below. % More generally, the third argument is read in vertical mode, with the % reference point at the lower left corner of the eps picture, whose % dimensions are contained in the dimen registers \drawx and \drawy. % This enables using TeX to generate the text that goes with the picture. % To request that the picture be widened to respect the added text, % examine and modify the dimen registers \ngap, \egap, \sgap, \wgap. % This is done automatically by the \figtext macro. % These macros rely on "epsf.tex" which is the lowest level interface % available for including encapsulated Postscript files in TeX documents. % Rather that manually reading the .eps file to compute the nominal size, % the \epsfbox macro is called twice, and two of its internal registers % are examined after the first call. A major change to epsf.tex (unlikely) % will require changes here. % add line \texturestrue to Textures version of epsf.tex \newif\iftextures \input epsf \newcount\figcount \figcount=0 \newcount\figxscale \newcount\figyscale \newcount\figxoffset \newcount\figyoffset \newbox\drawing \newcount\drawbp \newdimen\drawx \newdimen\drawy \newdimen\ngap \newdimen\sgap \newdimen\wgap \newdimen\egap \def\drawbox#1#2#3{\vbox{ \epsfgetbb{#2.eps} % computes \epsfllx \epsflly \epsfurx \epsfury \drawbp=\epsfurx \advance\drawbp by-\epsfllx\relax \multiply\drawbp by #1 \divide\drawbp by 100 \drawx=\drawbp bp \drawbp=\epsfury \advance\drawbp by-\epsflly\relax \multiply\drawbp by #1 \divide\drawbp by 100 \drawy=\drawbp bp \iftextures \figxscale=#1 \multiply\figxscale by 10 \setbox\drawing=\vbox to \drawy{\vfil \hbox to \drawx{\special{illustration #2.eps scaled \number\figxscale}\hfil}} \else %dvips \figxoffset=-\epsfllx \multiply\figxoffset by#1 \divide\figxoffset by100 \figyoffset=-\epsflly \multiply\figyoffset by#1 \divide\figyoffset by100 \setbox\drawing=\vbox to \drawy{\vfil \hbox to \drawx{\special{PSfile=#2.eps hoffset=\number\figxoffset\space voffset=\number\figyoffset\space hscale=#1 vscale=#1}\hfil}} \fi \setbox\drawing=\vbox{\offinterlineskip\box\drawing\kern 0pt} \ngap=0pt \sgap=0pt \wgap=0pt \egap=0pt \setbox0=\vbox{\offinterlineskip \box\drawing \ifgridlines\drawgrid\drawx\drawy\fi #3} \kern\ngap\hbox{\kern\wgap\box0\kern\egap}\kern\sgap}} \def\draw#1#2#3{ \setbox\drawing=\drawbox{#1}{#2}{#3} \global\advance\figcount by 1 \edef\numtoks{\number\figcount} \makeref{fig:#2}{Figure}{\numtoks} \goodbreak \midinsert \centerline{\ifgridlines\boxgrid\drawing\fi\box\drawing} \smallskip \vbox{\offinterlineskip \centerline{Figure~\number\figcount} \smash{\marginlabel{#2}}} \endinsert} \def\capdraw#1#2#3#4{ \setbox\drawing=\drawbox{#1}{#2}{#3} \global\advance\figcount by 1 \edef\numtoks{\number\figcount} \makeref{fig:#2}{Figure}{\numtoks} \goodbreak \midinsert \centerline{\ifgridlines\boxgrid\drawing\fi\box\drawing} \smallskip \vbox{\offinterlineskip \vskip 4pt \vbox{\centerline{\lineskip=3pt\small\noindent {\bf Figure~\number\figcount:} #4}} \smash{\marginlabel{fig:#2}}} \endinsert} \def\capdrawpair#1#2#3#4#5#6#7#8{ \goodbreak \midinsert \setbox\drawing=\drawbox{#1}{#2}{#3} \global\advance\figcount by 1 \edef\numtoks{\number\figcount} \makeref{fig:#2}{Figure}{\numtoks} \centerline{\ifgridlines\boxgrid\drawing\fi\box\drawing} \smallskip \vbox{\offinterlineskip \vskip 4pt \vbox{\lineskip=3pt\small\noindent {\bf Figure~\number\figcount:} #4} \smash{\marginlabel{fig:#2}}} \vskip 12pt \setbox\drawing=\drawbox{#5}{#6}{#7} \global\advance\figcount by 1 \edef\numtoks{\number\figcount} \makeref{fig:#6}{Figure}{\numtoks} \centerline{\ifgridlines\boxgrid\drawing\fi\box\drawing} \smallskip \vbox{\offinterlineskip \vskip 4pt \vbox{\lineskip=3pt\small\noindent {\bf Figure~\number\figcount:} #8} \smash{\marginlabel{fig:#6}}} \endinsert} \def\prevfigtoks{\edef\numtoks{\number\figcount}} \def\nextfigtoks{% \advance\figcount by 1% \edef\numtoks{\number\figcount}% \advance\figcount by -1} \def\prevfig{\prevfigtoks Figure~\numtoks} \def\nextfig{\nextfigtoks Figure~\numtoks} \def\prevfiglabel#1{\prevfigtoks\forward{#1}{Figure}{\numtoks}} \def\nextfiglabel#1{\nextfigtoks\forward{#1}{Figure}{\numtoks}} %=========================================================================% % figure text macros %=========================================================================% % \figtext{x1}{y1}{x2}{y2}{text} places "text" in a box, and overlays this % text onto an eps picture, when called as part of the third argument to % \draw, above. (x1,y1) are eps picture coordinates, from 0 to 1 % and relative to its lower left corner. (x2,y2) are text box % coordinates, from 0 to 1 and relative to its lower left corner. % Any of these coordinates may take values outside of the range 0 to 1. % These two points are attached, so text annotations survive changes of % scale of the eps picture. If text is placed extending outside the % eps picture, the box containing the eps picture is enlarged without % disturbing this coordinate system. % As an aid to placement, call the \gridlines macro to provide grid lines. % Crosshairs mark the reference point on each text box; minor ticks are % .002 units, and major ticks are .01 units. % To understand the motivation for the remaining macros, consider that % there are three separate units of measure impinging on a placement % of text over a picture: picture units, text units, and absolute units. % To survive picture rescaling and overall changes in the magnification of % the document, it is important to work out idioms that record placement % decisions in the correct units. % The macros \swtext, \nwtext, \setext, \netext, \wtext, \stext, \etext, % and \ntext refer to compass points on the perimeter of the text box, % replacing the (x2,y2) arguments to \figtext. Similarly, \ctext sets % (x2,y2) to the center of the text box. % The macros \swpad, \nwpad, \sepad, \nepad, \wpad, \spad, \epad, % and \npad place space in absolute TeX units around each box. The macro % \mimic gives the height of the first argument to its second % argument, so placements with north reference points come out aligned. % Thus, one idiom for using these macros would be: % \draw{70}{twistedfiber}{ % \setext{.594}{.288}{\sepad{3pt}{3pt}{$b^3c$}} % \ntext{.326}{0}{\npad{3pt}{\mimic{$b$}{$a$}}} % \ntext{.722}{0}{\npad{3pt}{$b$}} % } % As these macros operate in vertical mode, extra spaces are permitted. \newif\ifgridlines \newbox\figtbox \newbox\figgbox \newdimen\figtx \newdimen\figty \def\gridlines{\gridlinestrue} \newdimen\bwd \bwd=2sp % 2sp (1/32768") is smallest visible width for Textures \def\hline#1{\vbox{\smash{\hbox to #1{\leaders\hrule height \bwd\hfil}}}} \def\vline#1{\hbox to 0pt{% \hss\vbox to #1{\leaders\vrule width \bwd\vfil}\hss}} \def\clap#1{\hbox to 0pt{\hss#1\hss}} \def\vclap#1{\vbox to 0pt{\offinterlineskip\vss#1\vss}} \def\hstutter#1#2{\hbox{% \setbox0=\hbox{#1}% \hbox to #2\wd0{\leaders\box0\hfil}}} \def\vstutter#1#2{\vbox{ \setbox0=\vbox{\offinterlineskip #1} \dp0=0pt \vbox to #2\ht0{\leaders\box0\vfil}}} \def\crosshairs#1#2{ \dimen1=.002\drawx \dimen2=.002\drawy \ifdim\dimen1<\dimen2\dimen3\dimen1\else\dimen3\dimen2\fi \setbox1=\vclap{\vline{2\dimen3}} \setbox2=\clap{\hline{2\dimen3}} \setbox3=\hstutter{\kern\dimen1\box1}{4} \setbox4=\vstutter{\kern\dimen2\box2}{4} \setbox1=\vclap{\vline{4\dimen3}} \setbox2=\clap{\hline{4\dimen3}} \setbox5=\clap{\copy1\hstutter{\box3\kern\dimen1\box1}{6}} \setbox6=\vclap{\copy2\vstutter{\box4\kern\dimen2\box2}{6}} \setbox1=\vbox{\offinterlineskip\box5\box6} \smash{\vbox to #2{\hbox to #1{\hss\box1}\vss}}} \def\boxgrid#1{\rlap{\vbox{\offinterlineskip \setbox0=\hline{\wd#1} \setbox1=\vline{\ht#1} \smash{\vbox to \ht#1{\offinterlineskip\copy0\vfil\box0}} \smash{\vbox{\hbox to \wd#1{\copy1\hfil\box1}}}}}} \def\drawgrid#1#2{\vbox{\offinterlineskip \dimen0=\drawx \dimen1=\drawy \divide\dimen0 by 10 \divide\dimen1 by 10 \setbox0=\hline\drawx \setbox1=\vline\drawy \smash{\vbox{\offinterlineskip \copy0\vstutter{\kern\dimen1\box0}{10}}} \smash{\hbox{\copy1\hstutter{\kern\dimen0\box1}{10}}}}} \def\figtext#1#2#3#4#5{ \setbox\figtbox=\vbox{\hbox{#5}\kern 0pt} \figtx=-#3\wd\figtbox \figty=-#4\ht\figtbox \advance\figtx by #1\drawx \advance\figty by #2\drawy \dimen0=\figtx \advance\dimen0 by\wd\figtbox \advance\dimen0 by-\drawx \ifdim\dimen0>\egap\global\egap=\dimen0\fi \dimen0=\figty \advance\dimen0 by\ht\figtbox \advance\dimen0 by-\drawy \ifdim\dimen0>\ngap\global\ngap=\dimen0\fi \dimen0=-\figtx \ifdim\dimen0>\wgap\global\wgap=\dimen0\fi \dimen0=-\figty \ifdim\dimen0>\sgap\global\sgap=\dimen0\fi \smash{\rlap{\vbox{\offinterlineskip \hbox{\hbox to \figtx{}\ifgridlines\boxgrid\figtbox\fi\box\figtbox} \vbox to \figty{} \ifgridlines\crosshairs{#1\drawx}{#2\drawy}\fi \kern 0pt}}}} \def\nwtext#1#2#3{\figtext{#1}{#2}01{#3}} \def\netext#1#2#3{\figtext{#1}{#2}11{#3}} \def\swtext#1#2#3{\figtext{#1}{#2}00{#3}} \def\setext#1#2#3{\figtext{#1}{#2}10{#3}} \def\wtext#1#2#3{\figtext{#1}{#2}0{.5}{#3}} \def\etext#1#2#3{\figtext{#1}{#2}1{.5}{#3}} \def\ntext#1#2#3{\figtext{#1}{#2}{.5}1{#3}} \def\stext#1#2#3{\figtext{#1}{#2}{.5}0{#3}} \def\ctext#1#2#3{\figtext{#1}{#2}{.5}{.5}{#3}} % macros to add space to text on specified sides \def\hpad#1#2#3{\hbox{\kern #1\hbox{#3}\kern #2}} \def\vpad#1#2#3{\setbox0=\hbox{#3}\vbox{\kern #1\box0\kern #2}} \def\wpad#1#2{\hpad{#1}{0pt}{#2}} \def\epad#1#2{\hpad{0pt}{#1}{#2}} \def\npad#1#2{\vpad{#1}{0pt}{#2}} \def\spad#1#2{\vpad{0pt}{#1}{#2}} \def\nwpad#1#2#3{\npad{#1}{\wpad{#2}{#3}}} \def\nepad#1#2#3{\npad{#1}{\epad{#2}{#3}}} \def\swpad#1#2#3{\spad{#1}{\wpad{#2}{#3}}} \def\sepad#1#2#3{\spad{#1}{\epad{#2}{#3}}} % macros to kill depth of text string \def\killdp#1{\setbox0=\hbox{#1}\dp0=0pt\box0} % macro to give one text string the apparent height of another \def\mimic#1#2{\setbox1=\hbox{#1}\setbox2=\hbox{#2}\ht2=\ht1\box2} % macro to center one text string over another \def\stack#1#2#3{\vbox{\offinterlineskip \setbox2=\hbox{#2} \setbox3=\hbox{#3} \dimen0=\ifdim\wd2>\wd3\wd2\else\wd3\fi \hbox to \dimen0{\hss\box2\hss} \kern #1 \hbox to \dimen0{\hss\box3\hss}}} % macros to hide size of trailing exponents \def\sexp#1{\rlap{${}^{#1}$}} \def\hexp#1{% \setbox0=\hbox{${}^{#1}$}% \hbox to .5\wd0{\box0\hss}} \def\ssub#1{\rlap{${}_{#1}$}} \def\hsub#1{% \setbox0=\hbox{${}_{#1}$}% \hbox to .5\wd0{\box0\hss}} %=========================================================================% % macros for matrices and arrows %=========================================================================% % typical usage: % \rightarrowmat{2pt}{4pt}{d & bd \cr \!-c & 0 \cr 0 & -ac \cr} \def\bmatrix#1#2{{\left(\vcenter{\halign {&\kern#1\hfil$##\mathstrut$\kern#1\cr#2}}\right)}} \def\rightarrowmat#1#2#3{ \setbox1=\hbox{\small\kern#2$\bmatrix{#1}{#3}$\kern#2} \,\vbox{\offinterlineskip\hbox to\wd1{\hfil\copy1\hfil} \kern 3pt\hbox to\wd1{\rightarrowfill}}\,} \def\leftarrowmat#1#2#3{ \setbox1=\hbox{\small\kern#2$\bmatrix{#1}{#3}$\kern#2} \,\vbox{\offinterlineskip\hbox to\wd1{\hfil\copy1\hfil} \kern 3pt\hbox to\wd1{\leftarrowfill}}\,} \def\rightarrowbox#1#2{ \setbox1=\hbox{\kern#1\hbox{\small #2}\kern#1} \,\vbox{\offinterlineskip\hbox to\wd1{\hfil\copy1\hfil} \kern 3pt\hbox to\wd1{\rightarrowfill}}\,} \def\leftarrowbox#1#2{ \setbox1=\hbox{\kern#1\hbox{\small #2}\kern#1} \,\vbox{\offinterlineskip\hbox to\wd1{\hfil\copy1\hfil} \kern 3pt\hbox to\wd1{\leftarrowfill}}\,} %=========================================================================% % quire macros for preview mode and making booklets %=========================================================================% % \legalbooklet{20} makes a booklet from legal paper in landscape % orientation, where "20" is the page count. To preview, give a negative % pagecount. Either print using the legal duplex option on a modern laser % printer, or struggle to simulate this effect manually. Bind using a long % reach stapler. % \view is for previewing normal width pages (line breaks same as letter % output, but page breaks different) using Page Setup: A4 Landscape, then % Fit to Window followed by Fit to Width in Textures. % \twoup squeezes two pages onto letter paper in landscape mode, % suitable for printing. % each of these macros calls the file "quire.tex" % each of these only take effect within the first enclosing \input begin % thus, subsections can be always set to preview, while the top level % prints as a normal paper % \quiremacro is the metamacro which creates these macros. study and % modify a call to it, rather than modifying it directly. \def\quiremacro#1#2#3#4#5#6#7#8#9{ \expandafter\def\csname#1\endcsname##1{ \ifnum\srcdepth=1 \magnification=#2 \input quire \hsize=#3 \vsize=#4 \htotal=#5 \vtotal=#6 \shstaplewidth=#7 \shstaplelength=#8 % below computed from above \hoffset=\htotal \advance\hoffset by -\hsize \divide\hoffset by 2 \ifnum\vsize<\vtotal \voffset=\vtotal \advance\voffset by -\vsize \divide\voffset by 2 \fi \advance\voffset by #9 \shhtotal=2\htotal % below doesn't need to change \baselineskip=13pt \emergencystretch = 0.05\hsize \horigin=0.0truein \vorigin=0.0truein \shthickness=0pt \shoutline=0pt \shcrop=0pt \shvoffset=-1.0truein \ifnum##1>0\quire{#1}\else\qtwopages\fi \fi }} % name, magstep, % \hsize, \vsize, \htotal, \vtotal, \shstaplewidth, \shstaplelength % booklet macros \quiremacro{letterbooklet} % 4.79452 = 5.25/1.095 {1000}{4.79452truein}{7truein}{5.5truein}{8.5truein}{0.01pt}{0.66truein} {-.0625truein} \quiremacro{Afourbooklet} {1095}{5.25truein}{7truein}{421truept}{595truept}{0.01pt}{0.66truein} {-.0625truein} \quiremacro{legalbooklet} {1095}{5.25truein}{7truein}{7.0truein}{8.5truein}{0.01pt}{0.66truein} {-.0625truein} \quiremacro{twoupsub} % 895 = 1095*4.5/5.5 {895}{4.5truein}{7truein}{5.5truein}{8.5truein}{0pt}{0pt}{.0625truein} \def\twoup{\twoupsub{-1}} % use A4 landscape mode with \Afourview \quiremacro{Afourviewsub} % 5.0228311 = 5.5/1.095, 7.7625571 = 8.5/1.095 {1000}{5.0228311in}{7.7625571in}{421truept}{595truept}{0.1pt}{0.5\vtotal} {-.0625truein} \def\Afourview{\Afourviewsub{-1}} % use "Preview" portrait mode with \view % "Preview" is a custom Printer PDD entry with dimensions 922 by 666 \quiremacro{viewsub} {1095}{5.5truein}{8.5truein}{461truept}{666truept}{0.1pt}{0.5\vtotal} {-.125truein} \def\view{\viewsub{-1}} %=========================================================================% % timestamp (adapted from eplain.tex) %=========================================================================% \newcount\countA \newcount\countB \newcount\countC \def\monthname{\begingroup \ifcase\number\month \or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \endgroup} \def\dayname{\begingroup \countA=\number\day \countB=\number\year \advance\countA by 0 % adjust after each leap day \advance\countA by \ifcase\month\or 0\or 31\or 59\or 90\or 120\or 151\or 181\or 212\or 243\or 273\or 304\or 334\fi \advance\countB by -1995 \multiply\countB by 365 \advance\countA by \countB \countB=\countA \divide\countB by 7 \multiply\countB by 7 \advance\countA by -\countB \advance\countA by 1 \ifcase\countA\or Sunday\or Monday\or Tuesday\or Wednesday\or Thursday\or Friday\or Saturday\fi \endgroup} \def\timename{\begingroup \countA = \time \divide\countA by 60 \countB = \countA \countC = \time \multiply\countA by 60 \advance\countC by -\countA \ifnum\countC<10\toks1={0}\else\toks1={}\fi \ifnum\countB<12 \toks0={\sevenrm AM} \else\toks0={\sevenrm PM}\advance\countB by -12\fi \relax\ifnum\countB=0\countB=12\fi \hbox{\the\countB:\the\toks1 \the\countC \thinspace \the\toks0} \endgroup} \def\timestamp{\dayname, \the\day\ \monthname\ \the\year, \timename} %========================================================================== % choose page format %========================================================================== \print %\view %========================================================================== % macros (specific to this paper) %========================================================================== % author's comments \def\COMMENT#1\par{\bigskip\hrule\smallskip#1\smallskip\hrule\bigskip} % surround with $ $ if not already in math mode \def\enma#1{{\ifmmode#1\else$#1$\fi}} \def\mathbb#1{{\bbold #1}} \def\mathbf#1{{\bf #1}} % blackboard bold symbols \def\AA{\enma{\mathbb{A}}} \def\NN{\enma{\mathbb{N}}} \def\QQ{\enma{\mathbb{Q}}} \def\PP{\enma{\mathbb{P}}} \def\RR{\enma{\mathbb{R}}} \def\ZZ{\enma{\mathbb{Z}}} % caligraphic symbols \def\cAA{\enma{\cal A}} \def\cBB{\enma{\cal B}} \def\cEE{\enma{\cal E}} \def\cHH{\enma{\cal H}} \def\cFF{\enma{\cal F}} \def\cII{\enma{\cal I}} \def\cOO{\enma{\cal O}} % bold symbols \def\aa{\enma{\mathbf{a}}} \def\bb{\enma{\mathbf{b}}} \def\cc{\enma{\mathbf{c}}} \def\dd{\enma{\mathbf{d}}} \def\ee{\enma{\mathbf{e}}} \def\ff{\enma{\mathbf{f}}} \def\mm{\enma{\mathbf{m}}} \def\nn{\enma{\mathbf{n}}} \def\pp{\enma{\mathbf{p}}} \def\qq{\enma{\mathbf{q}}} \def\ss{\enma{\mathbf{s}}} \def\uu{\enma{\mathbf{u}}} \def\vv{\enma{\mathbf{v}}} \def\ww{\enma{\mathbf{w}}} \def\xx{\enma{\mathbf{x}}} \def\yy{\enma{\mathbf{y}}} \def\zz{\enma{\mathbf{z}}} \def\CC{\enma{\mathbf{C}}} \def\DD{\enma{\mathbf{D}}} \def\FF{\enma{\mathbf{F}}} \def\KK{\enma{\mathbf{K}}} \def\LL{\enma{\mathbf{L}}} \def\II{\enma{\mathbf{I}}} \def\zero{\enma{\mathbf{0}}} \def\mmod{\enma{\mathbf{mod}}} % misc \def\GL{\enma{\rm{GL}}} \def\Cl{\enma{\rm{Cl}}} \def\set#1{\enma{\{#1\}}} \def\setdef#1#2{\enma{\{\;#1\;\,|\allowbreak \;\,#2\;\}}} \def\ideal#1{\langle\,#1\,\rangle} \def\idealdef#1#2{\enma{\langle\;#1\;\, |\;\,#2\;\rangle}} \def\mtext#1{\;\,\allowbreak\hbox{#1}\allowbreak\;\,} \def\abs#1{\enma{\left| #1 \right|}} \def\floor#1{\enma{\lfloor #1 \rfloor}} \def\ceil#1{\enma{\lceil #1 \rceil}} % math keywords \def\conv{\mathop{\rm conv}\nolimits} \def\codim{\mathop{\rm cod}\nolimits} \def\hull{\mathop{\rm hull}\nolimits} \def\reg{\mathop{\rm reg}\nolimits} \def\im{\mathop{\rm im}\nolimits} \def\link{\mathop{\rm link}\nolimits} \def\rank{\mathop{\rm rank}\nolimits} \def\gldim{\mathop{\rm gl.dim}\nolimits} \def\Hom{\mathop{\rm Hom}\nolimits} \def\Ext{\mathop{\rm Ext}\nolimits} \def\Coh{\mathop{\rm Coh}\nolimits} % \def\ini{\mathop{\rm in}\nolimits} \def\fib{\mathop{\rm fib}\nolimits} %undefine\boxtimes \newsymbol\boxtimes1202 %========================================================================== % forward references needed in paper (specific to this paper) %========================================================================== \forward{introd}{Section}{1} \forward{unimodularity}{Section}{2} \forward{main}{Section}{3} \forward{mono}{Section}{4} \forward{examples}{Section}{5} \forward{diagonal}{Section}{6} \forward{beilinson}{Section}{7} \forward{fig:initial}{Figure}{4} \font\smallrm=cmcsc8 \headline={\ifodd\pageno \ifnum\pageno>1 \smallrm \hfil Syzygies of Unimodular Lawrence Ideals % running head for right-hand page is title in caps \hfil\folio \else\hfill\fi \else \smallrm \folio \hfill Dave Bayer, Sorin Popescu and Bernd Sturmfels % running head for left-hand page is authors in caps \hfill\fi} \footline={\hss} % footline is blank %\showlabels %\gridlines \overfullrule=0pt \hbox{} \smallskip \centerline{\titlefont Syzygies of Unimodular Lawrence Ideals} \bigskip \centerline{Dave Bayer \quad Sorin Popescu \quad Bernd Sturmfels} \bigskip \bigskip \noindent {Abstract:} {\sl Infinite hyperplane arrangements whose vertices form a lattice are studied from the point of view of commutative algebra. The quotient of such an arrangement modulo the lattice action represents the minimal free resolution of the associated binomial ideal, which defines a toric subvariety in a product of projective lines. Connections to graphic arrangements and to Beilinson's spectral sequence are explored.} \bigskip \section{introd} {Introduction} \noindent We are interested in the defining ideals of toric subvarieties in a product of projective lines $\, \PP^1 \times \PP^1 \times \cdots \times \PP^1$. Writing $(x_i:y_i)$ for the homogeneous coordinates of the $i$-th factor $\PP^1$, these are the binomial ideals in $2n$ variables of the following form: $$ J_L \quad = \quad \idealdef{\xx^\aa \yy^\bb - \xx^\bb \yy^\aa}{\aa-\bb \in L} \quad \subset \quad S \, = \, k[x_1,\ldots,x_n, \, y_1,\ldots,y_n], $$ where $L$ is a sublattice of $\ZZ^n$ and $k$ is a field. Here $\,\xx^\aa = \,x_1^{a_1} x_2^{a_2} \cdots x_n^{a_n} \,$ for $\aa = (a_1,\ldots,a_n) \in \NN^n$. Binomial ideals of the form $J_L$ are called {\it Lawrence ideals}. They provide the algebraic analogue to the Lawrence construction for convex polytopes [Zi, \S 6.6]. Lawrence polytopes enjoy remarkable rigidity properties, such as [Zi, Theorem 6.27]. On the algebraic side, rigidity of Lawrence ideals manifests itself in the following result, which appears in [Stu, Theorem 7.1]. Recall for part (d) that the {\it Graver basis} consists of all binomials $\xx^\aa \yy^\bb - \xx^\bb \yy^\aa$ in $J_L$ such that the only vector $\aa' - \bb' \in L \backslash \set{\zero}$ with $\zero \leq \aa'\leq \aa$ and $\zero \leq \bb' \leq \bb$ is $\aa - \bb$ itself. \proposition{lawrencegens} The following sets of binomials in a Lawrence ideal $J_L$ coincide: \item{\rm (a)} Any minimal set of binomial generators of $J_L$. \item{\rm (b)} Any reduced Gr\"obner basis for $J_L$. \item{\rm (c)} The universal Gr\"obner basis for $J_L$ (the union of all reduced Gr\"obner bases). \item{\rm (d)} The Graver basis for $J_L$. Cellular resolutions, as defined in [BS], provide a natural geometric framework for studying homological, algorithmic and combinatorial properties of monomial and binomial ideals. One instance is the {\it Scarf complex}, which gives the minimal resolution for generic monomial ideals [BPS] and generic lattice ideals [PS]. The {\it hull resolution} of [BS] generalizes the Scarf complex and provides a cellular resolution for arbitrary co-Artinian monomial modules; however, it need not be minimal. For lattice modules, the hull resolution is compatible with the lattice action and determines a cellular resolution of the corresponding lattice ideal [BS, Theorem 3.9]. In this paper we present a minimal cellular resolution, which happens to also coincide with the hull resolution, for a remarkable class of nongeneric lattice ideals. These are the unimodular Lawrence ideals $J_L$, which are characterized as follows: \theorem{unimod} For a sublattice $L$ of $\ZZ^n$ the following conditions are equivalent: \item{\rm (a)} The Lawrence ideal $J_L$ possesses an initial monomial ideal which is radical. \item{\rm (b)} Every initial monomial ideal of the Lawrence ideal $J_L$ is a radical ideal. \item{\rm (c)} Every minimal generator of $J_L$ is a difference of two squarefree monomials. \item{\rm (d)} The lattice $L$ is the image of an integer matrix $B$ with linearly independent columns, such that all maximal minors of $B$ lie in the set $\set{ 0, 1, -1}$. \item{\rm (e)} The lattice $L$ is the kernel of an integer matrix $A$ with linearly independent rows, such that all maximal minors of $A$ lie in $\set{ 0, m, -m}$ for some integer $m$. \item{\rm (f)} The quotient ring $S/J_L$ is a normal domain. \ref{unimod} is proved in \ref{unimodularity}. If any (and thus all) of these six equivalent conditions for $L$ holds, then we say that the Lawrence ideal $J_L$ is {\it unimodular\/}. A first example is the ideal of $2 \times 2$-minors of a $2 \times n$-matrix of indeterminates: $$ J_L \quad = \quad I_2 \pmatrix{ x_1 & x_2 & x_3 & \cdots & x_n \cr y_1 & y_2 & y_3 & \cdots & y_n \cr} \eqno (1.1) $$ Here $L$ is the kernel of $A = (\, 1 \, \,1 \, \,1 \, \, \cdots \, 1 \,)$, or the image of the matrix $B$ whose rows are $\ee_i - \ee_{i+1}$, $i\in\set{1,\ldots,n-1}$, the differences of consecutive unit vectors in $\RR^n$. The minimal resolution of (1.1) is an Eagon-Northcott complex, whose polyhedral model is the hypersimplicial complex of Gel'fand and MacPherson [BS, Ex.~3.15]. In \ref{unimodularity} we introduce an infinite periodic hyperplane arrangement $\cHH_L$ whose vertices are the elements of $L$. It is shown in \ref{main} that this arrangement supports the minimal free resolution of $J_L$, and coincides with the hull complex of $J_L$. In \ref{mono} we prove that this resolution is universal in the sense that it is stable under all Gr\"obner deformations. In particular, all initial ideals of $J_L$ have the same Betti numbers as $J_L$, and their minimal resolutions are also cellular. We also construct cellular resolutions for the monomial ideals defined by the fibers of $L$. In \ref{examples} we discuss Lawrence ideals associated with directed graphs and we present open combinatorial problems. In \ref{diagonal} we reinterpret Lawrence ideals in terms of the Audin-Cox homogeneous coordinate ring, and we generalize Beilinson's spectral sequence from projective space to unimodular toric varieties. The authors would like to thank the Mathematical Sciences Research Institute in Berkeley for its support while part of this paper was being written. All three authors were partially supported by the NSF during the preparation of this work. \section{unimodularity} {Unimodularity and an infinite hyperplane arrangement} \noindent In this section we establish some basic facts about unimodular lattices and their Lawrence ideals. We start out by proving the equivalences stated in \ref{introd}. \vskip .1cm \noindent {\bf Proof of \ref{unimod}.} A monomial ideal is radical if and only if it is generated by squarefree monomials. Clearly, the first term of a binomial $\,\xx^\aa \yy^\bb - \xx^\bb \yy^\aa\,$ is squarefree if and only if the second term is squarefree. The equivalence of (a), (b) and (c) follows directly from \ref{lawrencegens}. In an exact sequence of free abelian groups, $$ 0 \,\, \longrightarrow \,\,\ZZ^{n-d} \,\,\, {\buildrel B \over \longrightarrow } \,\,\, \ZZ^n \,\,\, {\buildrel A \over \longrightarrow } \,\,\, \ZZ^d, \eqno (2.1) $$ each $(n \! - \! d) \times (n \! - \!d)$-minor of $B$ is equal to the complementary $d \times d$-minor of $A$, up to a global constant $m$. Since the cokernel of $B$ is torsion free, the maximal minors of $B$ are integers with no common factor. This implies $m \in \ZZ$ and the equivalence of (d) and (e), for $\, L = \ker(A) = \im(B)$. The condition (e) is precisely the defining condition given in [Stu, \S 8, page 70] for a matrix $A$ to be unimodular. A matrix $A$ is unimodular if and only if its Lawrence lifting $\Lambda(A)$ is unimodular. Following [Stu, \S 7, page 55], the {\it Lawrence lifting\/} of the $d \times n$-matrix $A$ is obtained by appending the zero $d \times n$-matrix ${\mathbf 0}_{d,n}$ and two copies of the $n \times n$-identity matrix ${\mathbf I}_n$ as follows: $$ \Lambda(A) \quad = \quad \pmatrix{ A\hfill & {\mathbf 0}_{d,n}\hfill \cr {\mathbf I}_n\hfill & \, {\mathbf I}_n \hfill\cr}. $$ Hence the equivalence of (b) and (e) is a reformulation of [Stu, Remark 8.10]. The conjunction of (b) and (e) implies property (f), namely, the lattice $L$ is the kernel of an integer matrix if and only if $J_L$ is a prime ideal, and (b) implies normality by [Stu, Proposition 13.15]. To complete the proof, it suffices to show that (f) implies (c). Suppose that (c) is false, i.e., the ideal $J_L$ has a minimal generator $\,\xx^\aa \yy^\bb - \xx^\bb \yy^\aa\,$ whose terms are not squarefree. We may assume that this generator is a circuit, which means that it has minimal support. By setting all pairs of variables not appearing in this circuit to zero, we reduce to the case where $J_L$ is a principal ideal. But an affine binomial hypersurface is normal if and only if at least one of its monomials is squarefree. This completes our proof of \ref{unimod}. \Box Here is another, more invariant, formulation of the unimodularity condition. \proposition{coordproj} A lattice $L$ is unimodular if and only if, for every projection $\pi(L)$ of $L$ to a coordinate sublattice $\ZZ^r \subset \ZZ^n$, the quotient $\ZZ^r/\pi(L)$ is torsion free. \proof We will use the notation of the proof of \ref{unimod}. We first prove the ``if'' direction. Set $m = n-d$ and consider any nonsingular $m \times m$-submatrix $C$ of $B$. The image of $C$ in $\ZZ^m$ equals $\pi(L)$ for the corresponding coordinate projection $\pi : \ZZ^n \rightarrow \ZZ^m $. The finite abelian group $\ZZ^m/\pi(L)$ is torsion free if and only if it is zero. Hence $\pi(L) = \ZZ^m$ and we conclude that the determinant of the $m \times m$-submatrix of $B$ under consideration is either $1 $ or $-1$. For the ``only-if'' direction, suppose that $L$ is unimodular. Every coordinate projection $\pi(L)$ of $L$ is unimodular as well. This can be seen by choosing $A$ (resp.~$B$) to have appropriate unit vectors among its columns (resp.~rows). Thus it suffices to show that the group $\ZZ^n/L$ is torsion free. But this follows from the exact sequence (2.1), which identifies $\ZZ^n/L$ with the image of the matrix $A$, showing $\ZZ^n/L \simeq \ZZ^d$. \Box Let $L$ be any $m$-dimensional sublattice of $\ZZ^n$, and write $\RR L$ for the linear subspace of $\RR^n$ spanned by $L$. We denote by $\cHH_L$ the {\it affine hyperplane arrangement\/} in $\RR L$ obtained by intersecting $\RR L$ with all lattice translates of the coordinate hyperplanes in $\RR^n$. These are the hyperplanes $\set{x_i = j}$ for $\, 1\le i \le n $ and $ j \in \ZZ $. Thus $\cHH_L$ is an infinite $m$-dimensional hyperplane arrangement in the vector space $\RR L$. It is convenient to embed $\cHH_L$ as a hyperplane arrangement in the Euclidean space $\RR^m$ as follows. Let $B$ be an integer $n \times m$-matrix such that $L=\im(B)$ as in part (d) of \ref{unimod}. Write $\bb_i \in \ZZ^m$ for the $i$-th row vector of $B$. Then $\cHH_L$ is isomorphic to the infinite arrangement in $\RR^m$ consisting of the hyperplanes $$\, H_{ij} \,=\, \setdef{\xx \in \RR^m}{\bb_i \cdot \xx = j} \quad \hbox{ for all $i \in\set{1,2,\ldots,n}$ and all $j \in \ZZ$} .$$ \proposition{nonewvertices} Each lattice point in $L$ is a vertex of the affine hyperplane arrangement $\cHH_L$. There are no additional vertices in $\cHH_L$ if and only if $L$ is unimodular. \proof We identify $L$ with $\ZZ^m$ via the matrix $B$ and thus consider $\cHH_L$ as the hyperplane arrangement $\cHH_L=\set{H_{ij}}$ in $\RR^m$. The intersection point of $m$ linearly independent such hyperplanes, $$ H_{i_1 j_1} \, \cap \, H_{i_2 j_2} \, \cap \, \cdots \, \cap H_{i_m j_m} \quad = \quad \{ \xx \}, $$ is defined by the linear equations $\, \bb_{i_\nu} \cdot \xx \, = \,j_\nu \,$ for $\nu = 1,\ldots, m$. The point $\xx\in\RR^m$ has integer coordinates for all $j_1,\ldots,j_m \in \ZZ$ if and only if $\det(\bb_{i_1},\ldots,\bb_{i_m}) = \pm 1$. By part (d) in \ref{unimod}, this means that the lattice $L$ is unimodular. The first assertion holds because each $\xx \in \ZZ^m$ can be expressed as such an intersection. \Box From now on we assume that $L$ is a unimodular sublattice of $\ZZ^n$. Recall from [Stu, Proposition 8.11] that the Graver basis of the Lawrence ideal $J_L$ is given precisely by the circuits of the lattice $L$. (The {\it circuits\/} of $L$ are the primitive vectors in $L$ whose supports are minimal with respect to inclusion.) We view $\cHH_L$ as an infinite regular cell complex, equipped with an action by the abelian group $L$. \lemma{circuit} Two vertices $\aa$ and $\bb$ of the arrangement $\cHH_L$ are connected by an edge if and only if their difference $\aa - \bb$ is a circuit of the unimodular lattice $L$. \proof Since $L$ acts transitively on the vertices of $\cHH_L$ we may assume that $\bb = \zero$. Our assertion states that $\aa$ is a circuit if and only if $\{\zero,\aa\}$ forms an edge in $\cHH_L$. This holds because the circuits in the subspace $\RR L$ of $\RR^n$ are computed by the rule $$ H_{i_1 0} \, \cap \, H_{i_2 0} \, \cap \, \cdots \, \cap H_{i_{m-1} 0} \quad = \quad \RR \aa $$ for all possible increasing sequences of indices $\,1 \leq i_1 < i_2 < \cdots < i_{m-1} \leq n $. \Box In this section we have introduced a family of hyperplane arrangements $\cHH_L$ whose vertices form a lattice $L$. Such arrangements appear in many parts of the mathematical literature; for example, see [BLSWZ]. The group $L$ acts on the faces of the arrangement $\cHH_L$ with finitely many orbits, and we shall be interested in the quotient complex $\cHH_L/L$. Nontrivial examples will be presented in \ref{examples}. First we reinterpret $\cHH_L$ and $\cHH_L/L$ as minimal free resolutions in the sense of commutative algebra. This will be done in the next two sections by labeling the faces of $\cHH_L$ with Laurent monomials, following the general recipes in [BS]. Our warmup example (1.1) is the case where $L$ is spanned by the vectors $\ee_i - \ee_{i+1}$, $i\in\set{1,\ldots,n-1}$, the differences of the consecutive unit vectors in $\RR^n$. Thus $J_L$ is here the ideal of $2 \times 2$-minors, and $\cHH_L$ is the {\it hypersimplicial arrangement\/}. The quotient complex ${\cal H}_L/L$ has $n-1$ distinct maximal faces, called {\it hypersimplices\/}. The hypersimplicial arrangement for $n=3$ is depicted in the following figure: \capdraw{100}{hyper}{ \swtext{.075}{.875}{\swpad{2pt}{2pt}{${{y_1^2y_2^{}x_3^3} \over{x_1^2x_2^{}y_3^3}}$}} \swtext{.305}{.875}{\swpad{2pt}{2pt}{${{y_1^{}y_2^{}x_3^2} \over{x_1^{}x_2^{}y_3^2}}$}} \swtext{.530}{.875}{\swpad{2pt}{2pt}{${{y_2^{}x_3^{}}\over{x_2^{}y_3^{}}}$}} \swtext{.760}{.875}{\swpad{2pt}{2pt}{${{x_1^{}y_2^{}}\over{y_1^{}x_2^{}}}$}} \swtext{.990}{.875}{\swpad{2pt}{2pt}{${{x_1^2y_2^{}y_3^{}} \over{y_1^2x_2^{}x_3^{}}}$}} % \swtext{.190}{.505}{\swpad{2pt}{2pt}{${{y_1^2x_3^2}\over{x_1^2y_3^2}}$}} \swtext{.415}{.505}{\swpad{2pt}{2pt}{${{y_1^{}x_3^{}}\over{x_1^{}y_3^{}}}$}} \swtext{.635}{.505}{\swpad{4pt}{4pt}{$1$}} \swtext{.870}{.505}{\swpad{2pt}{2pt}{${{x_1^{}y_3^{}}\over{y_1^{}x_3^{}}}$}} % \swtext{.075}{.139}{\swpad{2pt}{2pt}{${{y_1^3x_2^{}x_3^2} \over{x_1^3y_2^{}y_3^2}}$}} \swtext{.305}{.139}{\swpad{2pt}{2pt}{${{y_1^2x_2^{}x_3^{}} \over{x_1^2y_2^{}y_3^{}}}$}} \swtext{.530}{.140}{\swpad{2pt}{3pt}{${{y_1^{}x_2^{}}\over{x_1^{}y_2^{}}}$}} \swtext{.760}{.139}{\swpad{2pt}{2pt}{${{x_2^{}y_3^{}}\over{y_2^{}x_3^{}}}$}} \swtext{.990}{.139}{\swpad{2pt}{2pt}{${{x_1^{}x_2^{}y_3^2} \over{y_1^{}y_2^{}x_3^2}}$}} } {The hypersimplicial arrangement for $n=3$.} % Its hypersimplices are up-triangles and down-triangles. Each vertex is labeled by a Laurent monomial. The finite quotient complex $\cHH_L/L$ is a torus, subdivided by one vertex, three edges and two $2$-cells, one up-triangle and one down-triangle. \section{main} {From hyperplane arrangements to minimal free resolutions} \noindent We fix a unimodular sublattice $L$ of dimension $m$ in $\ZZ^n$. The Laurent polynomial ring $T=k[x_1^{\pm 1},\ldots,x_n^{\pm 1}, \, y_1^{\pm 1},\ldots,y_n^{\pm1}]$ is a module over the polynomial ring $S = k[x_1,\ldots,x_n, \, y_1,\ldots,y_n]$. We consider the following monomial $S$-submodule of $T$: $$ M_L \quad := \quad S \cdot \setdef{\xx^\aa \yy^{-\aa}}{\aa \in L} \quad \subset \quad T.$$ Each lattice point $\aa = (a_1,\ldots,a_n)$ in $L$ is a vertex of the arrangement $\cHH_L$, and we label that vertex of $\cHH_L$ with the corresponding generator of $M_L$, namely, $$ \xx^\aa \yy^{-\aa} \quad := \quad x_1^{a_1} x_2^{a_2} \cdots x_n^{a_n} y_1^{-a_1} y_2^{-a_2} \cdots y_n^{-a_n}. $$ Each face $F$ of $\cHH_L$ is then labeled by the least common multiple $m_F$ of the labels of its vertices. (The {\it least common multiple\/} of a set of Laurent monomials is the Laurent monomial whose exponents are the coordinatewise maxima of the given exponents.) The labeled cell complex $\cHH_L$ defines a complex of free $\ZZ^{2n}$-graded $S$-modules $$\FF_{\cHH_L} \,\, = \,\,\bigoplus_{F\in\cHH_L \atop F\ne\emptyset} S( -m_F ),$$ where the summand $S(-m_F)$ has homological degree $\dim(F)$. The differential of the complex $\FF_{\cHH_L}$ is the homogenized differential of the cell complex $\cHH_L$, defined by $$\partial(F) \quad = \,\sum_{F'\subset F \atop \codim(F',F)=1} \epsilon(F,F') \cdot {m_F \over m_{F'}}\cdot F',\qquad \hbox{for faces} \quad F \,\,\,\hbox{of} \,\,\, \cHH_L.$$ Here $\epsilon(F,F')$ is either $+1$ or $-1$, indicating the orientation of $F'$ in the boundary of $F$. See [BS, \S1] for details on this construction. The complex $(\FF_{\cHH_L},\partial)$ is not $S$-finite, but has finite length $m=\rank(L)$. As an example consider the complex of free $k[x_1,x_2,x_3,y_1,y_2,y_3]$-modules defined by the hypersimplicial arrangement in \ref{fig:hyper}. The edge $E$ connecting the module generators $\,m_1 = {x_2 y_3 \over y_2 x_3}\,$ and $\,m_2 = {x_1 y_3 \over y_1 x_3}\,$ is labeled by their least common multiple, which is the Laurent monomial $\,m_E = {x_1 x_2 y_3 \over x_3}$. This edge $E$ represents the first syzygy $( x_1 y_2) \cdot m_1 - (y_1 x_2) \cdot m_2$ of the monomial submodule $M_L$. \theorem{mainthm} The complex $(\FF_{\cHH_L},\partial)$ is a minimal $\ZZ^{2n}$-graded free $S$-resolution of the lattice module $M_L$. \proof The complex $(\FF_{\cHH_L},\partial)$ consists of free $S$-modules and is clearly $\ZZ^{2n}$-graded. To show that it is a resolution, we apply the exactness criterion in [BS, Proposition 1.2] to the labeled cell complex $X = \cHH_L$. For any $(\aa,\bb) \in \ZZ^{2n}$ we consider the subcomplex $\,X_{\leq (\aa,\bb)} \,$ consisting of all faces $F$ of the arrangement $X$ whose label $\,m_F = \xx^\cc \yy^\dd \,$ satisfies the coordinatewise inequalities $\,\cc \leq \aa \,$ and $\,\dd \leq \bb $. We shall prove that $X_{\leq (\aa,\bb)}$ is contractible, by identifying this subcomplex with a convex polytope in $\RR L$. For instance, the marked pentagon in \ref{fig:hyper} is $\,X_{\leq (2,1,1,\, 1,1,1)}$. Our labeled hyperplane arrangement can be described as follows. For $\uu \in \RR^n$ we write $\,(\ceil{\uu}, \floor{\uu})\,$ for the vector in $\ZZ^{2n}$ obtained by rounding up and down each coordinate of $\uu$. For instance, if $\uu = ( -2/5,-1, 7/5)\,$ then $\,(\ceil{\uu}, \floor{\uu}) = (0,-1,2 , -1, -1,1) $. Two vectors $\uu$ and $\vv$ in the subspace $\RR L$ of $\,\RR^n \,$ are called ${\it equivalent\/}$ if $\,(\ceil{\uu}, \floor{\uu}) \, = \,(\ceil{\vv}, \floor{\vv}) $. The resulting equivalence classes on $\RR L$ are the (relatively open) faces of $X = {\cal H}_L$. The face $F$ containing $\uu \in \RR L$ is labeled by the vector $\, \,(\ceil{\uu}, -\floor{\uu})$, or by the corresponding Laurent monomial $m_F=\xx^{\ceil{\uu}} \yy^{-\floor{\uu}}$. For $\aa,\bb \in \ZZ^n$ we consider the following subset of the subspace $\,\RR L \,$ in $\, \RR^n$: $$ \setdef{\uu \in \RR L}{ \ceil{\uu} \leq \aa \,\,\, \hbox{and} \,\,\, -\floor{\uu} \leq \bb} \quad = \quad \setdef{ \uu \in \RR L}{ - \bb \leq \uu \leq \aa}. \eqno (3.1{\rm a}) $$ This is a convex polytope with facet hyperplanes taken from $\cHH_L$. By the construction in the previous paragraph, the complex $\,X_{\leq (\aa,\bb)} \,$ is a polyhedral subdivision of the convex polytope identified in (3.1a). Therefore $\,X_{\leq (\aa,\bb)} \,$ is contractible, and, by [BS, Proposition 1.2], the complex $(\FF_{\cHH_L},\partial)$ is exact over $S$. It follows from the description of the labeled complex $ X = \cHH_L$ in the second-to-last paragraph that distinct faces $F$ and $F'$ of $X$ have distinct labels $m_F \not= m_{F'}$. This shows that the resolution $(\FF_{\cHH_L},\partial)$ is minimal (compare [BS, Remark 1.4]). \Box \corollary{ZeroOne} $\!\! \!\! $ The $\ZZ^{2n}$-graded Betti numbers of $\,M_L\,$ are $0$ or $1$. \Box %\vskip .1cm We retain the following identification for the rest of the paper: $$ X_{\leq (\aa,\bb)} \quad = \quad \setdef{\uu \in \RR L}{- \bb \leq \uu \leq \aa} \qquad \hbox{for} \quad \aa ,\bb \in \ZZ^n. \eqno (3.1{\rm b}) $$ In the next section we shall further identify these polytopes with the {\it fibers\/} of the Lawrence ideal $J_L$, i.e., with the congruence classes modulo $J_L$ of monomials in $S$. Note that the polytope $\, X_{\leq (\aa,\bb)} \,$ is itself a (closed) face of the arrangement $X = \cHH_L$ if and only if there exists a vector $\uu \in \RR L $ such that $\,(\aa,\bb) = (\ceil{\uu}, -\floor{\uu})$. Suppose that this holds. Then each coordinate of $\,\aa + \bb \,$ is either $0$ or $1$, and, taking any vertex of $\,X_{\leq (\aa,\bb)}$, we get a lattice point $\vv \in L$ with $\,-\bb \leq \vv \leq \aa $. By translating our face with the lattice vector $\vv$, we obtain now the following conclusion: \proposition{quotientL} Modulo the action by the unimodular lattice $L$, each face of ${\cal H}_L$ has the form $\,X_{\leq (\aa,\bb)}$, where $\aa \in \set{0,1}^n$ and $\bb\in \set{0,1}^n$ have disjoint support. \Box We next consider the quotient complex $\cHH_L/L$, which is formally defined as the face poset of $\cHH_L$ modulo the action by the lattice $L$. \ref{quotientL} implies: \corollary{isFinite} The number of faces of the quotient complex $\cHH_L/L$ is finite. \Box We apply the algebraic quotient construction in [BS, Section 3] to this quotient complex. The labeling of the $\ZZ^{2n}$-graded cell complex $X = {\cal H}_L$ is consistent with the action by the following lattice which is canonically isomorphic to $L$, $$ \Lambda(L) \quad := \quad \setdef{ (\uu,-\uu) \in \ZZ^{2n}}{ \uu \in L}. \eqno (3.2) $$ Following [BS, Lemma 3.5], the complex $(\FF_{\cHH_L},\partial)$ has the structure of a complex of $\ZZ^{2n}$-graded free modules over the group algebra $\, S[L]$. The rank of $\FF_{\cHH_L}$ over $S[L]$ equals the number of faces of $\cHH/L$, which is finite by \ref{isFinite}. Now the functor in [BS, Theorem 3.2] defines an equivalence of categories between the category of $\ZZ^{2n}$-graded $S[L]$-modules and the category of $\ZZ^{2n}/\Lambda(L)$-graded $S$-modules. Applying this functor to the $S[L]$-complex $(\FF_{\cHH_L},\partial)$ we obtain the cellular quotient complex $\,(\FF_{\cHH_L/L},\partial)\,$ which is a complex of $\ZZ^{2n}/\Lambda(L)$-graded free $S$-modules. From \ref{mainthm} and [BS, Corollary 3.7] we conclude that $\,(\FF_{\cHH_L/L},\partial)\,$ is the minimal free resolution of $S/J_L$ over $S$. This concludes the proof of the following theorem. \theorem{unimod-toric-syz} The quotient complex $\cHH_L/L$ of the hyperplane arrangement modulo $L$ supports the minimal $S$-free resolution of the unimodular Lawrence ideal $J_L$. \Box \corollary{ZeroOneagain} The minimal free resolution of the unimodular Lawrence ideal $J_L$ is independent of the characteristic of the base field $k$. The number of minimal $i$-th syzygies of $S/J_L$ equals the number of $i$-dimensional faces of the quotient complex $\cHH_L/L$, and the Betti numbers of $S/J_L$ in the $\ZZ^{2n}/\Lambda(L)$-grading are all $0$ or $1$. \Box %\vskip .1cm To write down the matrices in the minimal cellular resolution $\,(\FF_{\cHH_L/L},\partial)\,$ of a unimodular Lawrence ideal $J_L$, one must select a fundamental domain of $\cHH_L$ modulo $L$ and identify the cover relation in the poset of faces of $\cHH_L/L$. In higher dimensions it is convenient to use the monomial ideals in \ref{mono} for that purpose, but in dimensions $2$ and $3$ we can do the identifications directly on the picture. For instance, if $\,L = \ker(1 \,\, 1 \,\, 1 )\,$ and thus $J_L$ is the ideal of $2 \times 2$-minors of a generic $2 \times 3$-matrix, then the resolution $\,(\FF_{\cHH_L/L},\partial)\,$ is supported by the hypersimplicial arrangement in $\RR^2$ from \ref{fig:hyper}. Here the quotient complex $\cHH_L/L$ consists of one vertex, three edges and two triangles, which are glued to form a torus. Hence $S/J_L$ has one generator, three first syzygies and two second syzygies: $$ 0 \longrightarrow S^2 \rightarrowmat{2pt}{4pt}{x_1 & y_1 \cr x_2 & y_2 \cr x_3 & y_3 \cr} S^3 \rightarrowmat{2pt}{4pt}{x_2 y_3 \! - \! x_3 y_2 & x_3 y_1 \! - \! x_1 y_3 & x_1 y_2 \! - \! x_2 y_1 \cr} S \longrightarrow S/J_L\longrightarrow 0 $$ \vskip .1cm We now show that the minimal cellular resolution of $S/J_L$ equals the hull resolution introduced in [BS, Section 3]. For $\aa$, $\bb\in\ZZ^n$ and $t > 0$, we write $$(t^\aa,t^\bb) \quad = \quad (t^{a_1},\ldots,t^{a_n},t^{b_1},\ldots,t^{b_n}) \, \in \, \RR^{2n}$$ Recall that $\hull(M_L)$ is the complex of bounded faces of the polyhedron $P_t$ for large $t$, where $P_t$ is the convex hull of $\setdef{(t^\aa,t^\bb)}{\xx^\aa \yy^\bb \in M_L}\subset \RR^{2n}$. The vertices of $P_t$ correspond to the minimal generators of $M_L$, and the {\it hull resolution\/} of $S/J_L$ is the cellular resolution supported on $\hull(M_L)/L$. We shall use the following lemma. \lemma{hardanalysis} Let $\set{a, b}$ and $\set{c,d}$ be pairs of integers so $a+b=c+d$, and let $t>0$, $t\ne 1$. If $\abs{a-b} = \abs{c-d}$, then $t^a + t^b = t^c + t^d$. If $\abs{a-b} > \abs{c-d}$, then $t^a + t^b > t^c + t^d$. \proof If $\abs{a-b} = \abs{c-d}$, then $\set{a,b} = \set{c,d}$ as multisets. If $\abs{a-b} > \abs{c-d}$, suppose to be definite that $a > c \ge d > b$. In view of $a-c=d-b$, we have $\, t^a-t^c-t^d+t^b \;=\; t^c(t^{a-c}-1)-t^b(t^{d-b}-1) \;=\; t^b(t^{c-b}-1)(t^{a-c}-1) \;>\; 0 $. \Box \theorem{hullthm} The hull resolution of $S/J_L$ agrees with the minimal free resolution. \proof We show that $\hull(M_L)$ and $\cHH_L$ agree as cell complexes, using the fact that $\hull(M_L)$ consists of those faces of $P_t$ supported by strictly positive inner normals. Let $F$ be a face of $\cHH_L$. Then $F = X_{\leq (\ceil{\uu}, -\floor{\uu})}$ for some $\uu \in \RR L$. In other words, the vertices of $F$ are precisely those elements $\aa\in L$ so $\floor{u_i} \le a_i \le \ceil{u_i}$ for each $i$. Let $\vv=(t^{-\ceil{\uu}}, t^{\floor{\uu}})$. If $\aa$ is a vertex of $F$, then $\vv\cdot(t^\aa,t^{-\aa}) = n + m +(n-m)t^{-1}$, where $m$ is the number of coordinates in which $\uu$ is an integer. Suppose that $\bb\in L$ is not a vertex of $F$, and let $t>1$. Write $\vv_i = (t^{-\ceil{u_i}}, t^{\floor{u_i}})$, so $\vv\cdot(t^\aa,t^{-\aa}) = \sum_{i=1}^n \vv_i \cdot (t^{a_i},t^{-a_i})$. For each $i$ we have $\vv_i\cdot (t^{b_i},t^{-b_i}) \ge \vv_i\cdot (t^{a_i},t^{-a_i})$, with equality if and only if $\floor{u_i} \le b_i \le \ceil{u_i}$. This follows by applying \ref{hardanalysis} to the pairs $\set{b_i-\ceil{u_i}, \floor{u_i}-b_i}$ and $\set{a_i-\ceil{u_i}, \floor{u_i}-a_i}$. Thus $\vv\cdot(t^\bb,t^{-\bb}) > \vv\cdot(t^\aa,t^{-\aa})$, so $\vv$ supports $F$ as a face of $\hull(M_L)$. Let $\aa$ and $\bb$ be two vertices of $\cHH_L$ which do not belong to a common face of $\cHH_L$, and let $F = X_{\leq (\ceil{\uu}, -\floor{\uu})}$ be the face of $\cHH_L$ determined by $\uu=(\aa+\bb)/2$. Then $\abs{a_i-b_i} \ge 2$ for some $i$, for otherwise $\aa$ and $\bb$ would both belong to $F$ by \ref{quotientL}. Let $\cc$ be any vertex of $F$, and let $\dd=\aa+\bb-\cc$. Note that for each $j$, $\set{c_j, d_j}=\set{\floor{u_j}, \ceil{u_j}}$ as multisets, so $\abs{c_j-d_j}\le 1$. Then $\aa+\bb=\cc+\dd$, and $\abs{a_j-b_j} \ge \abs{c_j-d_j}$ for each $j$, with strict inequality for $j=i$. Let $t>1$. Applying \ref{hardanalysis} to the pairs $\set{a_j,b_j}$ and $\set{c_j,d_j}$, $$t^{a_j}+t^{b_j} \ge t^{c_j}+t^{d_j} \quad\mtext{and}\quad t^{-a_j}+t^{-b_j} \ge t^{-c_j}+t^{-d_j}$$ for each $j$, with strict inequalities for $j=i$. Let $\pp$ be the midpoint of the line segment from $(t^\aa,t^{-\aa})$ to $(t^\bb,t^{-\bb})$, and let $\qq$ be the midpoint of the line segment from $(t^\cc,t^{-\cc})$ to $(t^\dd,t^{-\dd})$. We have shown that $\pp-\qq$ is a nonzero, nonnegative vector. Therefore, the point $\pp$ cannot lie on any face of $\hull(M_L)$, because $\vv\cdot\pp > \vv\cdot\qq$ for any strictly positive vector $\vv$. Thus $\aa$ and $\bb$ cannot belong to a common face of $\hull(M_L)$. We conclude that the cell complexes $\hull(M_L)$ and $\cHH_L$ are equal. \Box \section{mono} {Fiber monomial ideals and initial monomial ideals} \noindent With any lattice ideal in a polynomial ring one can associate two families of monomial ideals. First, there are the {\it initial monomial ideals\/}, with respect to various term orders. Their minimal free resolutions can always be lifted, by Schreyer's construction in Gr\"obner basis theory, to a (possibly nonminimal) resolution of the lattice ideal. Second, we have the {\it fiber monomial ideals\/}, which are generated by the fibers, that is, the equivalence classes of monomials modulo the lattice ideal [PS, Section 2]. It follows from the results in [BS, Section 3] that the resolution of the lattice ideal is always determined by the resolution of a large enough fiber ideal. In this section we make these results precise for the unimodular Lawrence case. Fix a unimodular sublattice $L\subset\ZZ^n$. Two monomials $m$ and $m'$ in $S = k[x_1,\ldots,x_n,y_1,\ldots,y_n]$ are considered {\it equivalent\/} if $m - m'$ lies in the unimodular Lawrence ideal $J_L$. The equivalence classes are finite and called the {\it fibers\/} of $J_L$. For given $\aa,\bb \in \NN^n$, let $\fib(\aa,\bb)$ denote the fiber of the monomial $\,\xx^\aa \yy^\bb$. We shall identify the fibers with the lattice points in the polytopes of the form (3.1b): \lemma{lawrence-fibers} Let $X = \cHH_L$ be the labeled cell complex introduced in \ref{main}. Then the map $\,\,\, \phi \, : \, {\fib}(\aa,\bb) \rightarrow X_{\leq (\aa,\bb)} \, \cap \, L \, , \,\, \xx^\cc \yy^\dd \mapsto \aa - \cc \,\,\,$ is a bijection. \proof A monomial $\,\xx^\cc \yy^\dd \,$ in $S$ lies in ${\fib}(\aa,\bb)$ if and only if the vector $\,(\aa - \cc, \bb - \dd)\in\ZZ^{2n}$ lies in the lattice $\Lambda(L)$ defined in (3.2). The latter condition means that $\, \uu = \aa - \cc \,$ lies in $L$ and $\, \bb + \uu = \dd $. Thus $\,\uu \,$ is a vertex of $X = \cHH_L$ and its label $$ \xx^\uu \yy^{-\uu} \quad = \quad \xx^{\aa - \cc} \yy^{\bb - \dd} \quad \hbox{divides} \quad \xx^{\aa} \yy^{\bb } , $$ which shows that $\uu$ is actually a vertex of $X_{\leq (\aa,\bb)}$. Hence the map $\phi$ is well-defined. To see that it is a bijection we note that the inverse map is given by $\,\phi^{-1} (\uu) \,=\, \xx^{ \aa - \uu} \yy^{ \bb + \uu} $. This map is an analogue of the bijection in [PS, (2.1)]. \Box Let $\, \langle {\fib}(\aa,\bb) \rangle \,$ denote the ideal in $S$ generated by all monomials in the fiber of $\xx^\aa \yy^\bb$. The vertices of the cell complex $\, X_{\leq (\aa,\bb)} \, $ (considered as a subcomplex of $\cHH_L$) are labeled with certain Laurent monomials of the form $\,\xx^{\uu} \yy^{-\uu}$, with $\uu \in L$. Consider their preimages under the bijection $\phi$, and let $\, Y_{(\aa,\bb)} \, $ denote the same cell complex as $\, X_{\leq (\aa,\bb)} \, $ but with the vertices labeled by the monomials in ${\fib}(\aa,\bb)$. Thus the vertex with label $\,\xx^{\uu} \yy^{-\uu}\,$ in $ \,X_{\leq (\aa,\bb)} \, $ becomes the vertex with label $\,\xx^{\aa-\uu} \yy^{\bb+\uu} \,$ in $\,Y_{(\aa,\bb)}$. The labeled cell complex $\,Y_{(\aa,\bb)}\,$ gives rise to a $\ZZ^{2n}$-graded complex of free $S$-modules as in [BS, Section 1]. This complex is always exact and minimal: \theorem{fiberRes} Let $L$ be a unimodular sublattice of $\ZZ^n$ and $\aa,\bb \in \NN^n$. The labeled cell complex $Y_{(\aa,\bb)}$ defines a minimal free resolution of the monomial ideal $\, \langle {\fib}(\aa,\bb) \rangle $. \proof For any $\cc,\dd \in \NN^n$ we can make the following identification of cell complexes $$ (Y_{(\aa,\bb)})_{\leq (\cc,\dd)} \quad = \quad X_{\leq ( {\min}\{\bb,\cc-\aa\}, \, {\min}\{\aa,\dd-\bb \} )}, \eqno (4.1) $$ where ``$\min{}\!$'' refers to the coordinatewise minimum, and the labels are shifted appropriately. To see that (4.1) holds, one identifies both sides with the convex polytope $$ \setdef{ \uu \in {\RR L}} {(\zero,\zero) \leq (\aa+ \uu, \bb-\uu) \leq (\cc,\dd)}, $$ together with its natural subdivision. We conclude that the relevant subcomplexes for the exactness criterion in [BS, Proposition 1.2] are all contractible. As above, in the proof of \ref{mainthm}, distinct faces of $\,Y_{(\aa,\bb)}\,$ have distinct labels. Therefore $\,Y_{(\aa,\bb)} \,$ supports a minimal cellular resolution of the fiber ideal $\,\langle {\fib}(\aa,\bb) \rangle $. \Box \example{magicmatrices} Let $L$ be the corank $1$ lattice of all vectors in $\ZZ^n$ with coordinate sum zero. As discussed at the end of \ref{unimodularity}, $X = \cHH_L$ is the hypersimplicial arrangement in $\RR^{n-1}$, while $J_L$ is the ideal of $2 \times 2$-minors of the matrix (1.1). Let $M$ be the ideal generated by the monomials $\, x_1^{u_1} x_2^{u_2} \cdots x_n^{u_n} y_1^{v_1} y_2^{v_2} \cdots y_n^{v_n} \,$ where $$ \pmatrix{ u_1 & u_2 & \cdots & u_n \cr v_1 & v_2 & \cdots & v_n \cr} \eqno (4.2) $$ runs over all nonnegative integer matrices having the same row sums and the same column sums. Then the minimal free resolution of $M$ is cellular and supported by a suitably relabeled subcomplex $Y_{(\aa,\bb)}$ of the hypersimplicial arrangement $X = \cHH_L$. For instance, let $n=3$ and consider the following monomial ideal $$ \eqalign{ M \quad = \quad \langle \; x_2^2 x_3^2 y_1^3 , \; x_1 x_2 x_3^2 y_1^2 y_2 , \; & x_1^2 x_3^2 y_1 y_2^2 , \; x_1 x_2^2 x_3 y_1^2 y_3 , \; \cr & x_1^2 x_2 x_3 y_1 y_2 y_3 , \; x_1^3 x_3 y_2^2 y_3 , \; x_1^2 x_2^2 y_1 y_3^2 , \; x_1^3 x_2 y_2 y_3^2 \; \rangle \cr} $$ Here the matrices (4.2) have row sums $4,3$ and column sums $3,2,2$. The minimal free resolution of $M$ is supported on the complex $Y_{(2,1,1,\, 1,1,1)}$ which is a pentagon: % \capdraw{100}{pentagon}{ \stext{.212}{1}{\spad{3pt}{\small${x_1^2x_2^2y_1^{}y_3^2}$}} \stext{.595}{1}{\spad{3pt}{\small${x_1^{}x_2^2x_3^{}y_1^2y_3^{}}$}} \stext{.979}{1}{\spad{3pt}{\small${x_2^2x_3^2y_1^3}$}} % \etext{.000}{.500}{\epad{4pt}{\small${x_1^3x_2^{}y_2^{}y_3^2}$}} \stext{.403}{.538}{\spad{3pt}{\small${x_1^2x_2^{}x_3^{}y_1^{}y_2^{}y_3^{}}$}} \wtext{.8080}{.500}{\wpad{4pt}{\small${x_1^{}x_2^{}x_3^2y_1^2y_2^{}}$}} % \etext{.195}{.036}{\epad{4pt}{\small${x_1^3x_3^{}y_2^2y_3^{}}$}} \wtext{.614}{.036}{\wpad{4pt}{\small${x_1^2x_3^2y_1^{}y_2^2}$}} % }{Minimal free resolution of a fiber in the hypersimplicial arrangement.} % Let $\prec$ be any term order on the polynomial ring $\, S = k[x_1,\ldots,x_n,y_1,\ldots,y_n]$. Consider the initial monomial ideal $\,{\ini}_\prec(J_L)\,$ of the unimodular Lawrence ideal $J_L$. We know from \ref{unimod} that $\ini_\prec(J_L)$ is a squarefree monomial ideal. We shall describe a minimal cellular free resolution of $\,{\ini}_\prec(J_L)\,$ and show that it has the same Betti numbers as the resolution $\,(\FF_{\cHH_L/L},\partial)\,$ of the Lawrence ideal $J_L$. Let $\cHH^0_L$ be the set of all faces of the infinite hyperplane arrangement $\cHH_L$ which contain the origin $\zero \in L$. This is a finite cell complex which we identify with the central hyperplane arrangement in $\RR L $ given by the $n$ coordinate hyperplanes $x_i = 0$. The faces of ${\cHH}^0_L$ are cones in $\RR L$ with their apex at the origin. Under the embedding of $\cHH_L$ in $ \RR^m$ given prior to \ref{nonewvertices}, the complex ${\cHH}^0_L$ becomes the central arrangement defined by the hyperplanes $\,H_{10}, H_{20},\ldots, H_{n0}\,$ in $\RR^m$. For instance, in the example of \ref{fig:hyper}, $\cHH^0_L$ consists of one $0$-face, six $1$-faces and six $2$-faces. The term order $\prec$ extends uniquely to a total order on all Laurent monomials $\xx^{\uu} \yy^{-\uu}$ and hence on all vertices $\uu$ of $\cHH_L$. A positive-dimensional cone $F$ in the central arrangement ${\cHH}^0_L$ is called {\it $\prec$-positive\/} if all nonzero vertices $\uu \in L $ of the face $F$ satisfy the inequality $\,1 \prec \xx^{\uu} \yy^{-\uu} $. We can represent the term order $\prec$ by a generic hyperplane $H_\prec$ not containing the origin in $\RR L$, such that the $\prec$-positive cones of ${\cHH}^0_L$ are precisely those cones which have bounded, nonempty intersection with $H_\prec$. If $F$ is an $i$-dimensional $\prec$-positive cone in $\cHH^0_L$ then $ \, F \,\cap \,H_\prec\,$ is an $(i\! - \! 1)$-dimensional convex polytope. We write ${\ini}_\prec(\cHH_L)$ for the $(m-1)$-dimensional labeled cell complex consisting of those (bounded) polytopes $ \, F \,\cap \,H_\prec$. Here $ F \,\cap \,H_\prec $ inherits the label $m_F$ from the face $F$ of $\cHH_L$. Note that $m_F$ is a monomial in $S$ since $\,\zero \in F$. % \capdraw{100}{top}{ \wtext{.507}{.950}{\wpad{3pt}{$H_\prec$}} % \swtext{.290}{.780}{\swpad{2pt}{2pt}{${{y_2x_3}\over{x_2y_3}}$}} \swtext{.595}{.780}{\swpad{2pt}{2pt}{${{x_1y_2}\over{y_1x_2}}$}} % \swtext{.140}{.460}{\swpad{2pt}{2pt}{${{y_1x_3}\over{x_1y_3}}$}} \stext{.405}{.480}{\spad{4pt}{$1$}} \swtext{.745}{.460}{\swpad{2pt}{2pt}{${{x_1y_3}\over{y_1x_3}}$}} % \swtext{.295}{.135}{\swpad{2pt}{2pt}{${{y_1x_2}\over{x_1y_2}}$}} \swtext{.595}{.135}{\swpad{2pt}{2pt}{${{x_2y_3}\over{y_2x_3}}$}} % \wtext{.995}{.666}{\wpad{3pt}{$x_1y_2$}} \wtext{.995}{.456}{\wpad{3pt}{$x_1y_3$}} \wtext{.995}{.230}{\wpad{3pt}{$x_2y_3$}} \ntext{1.0}{.140}{\npad{3pt}{${\ini}_\prec(\cHH_L)$}} % }{Initial monomial ideal of a unimodular Lawrence ideal.} % \ref{fig:top} shows such a hyperplane $H_\prec$ and the resulting complex $ \ini_\prec(\cHH_L) $ arising from the hypersimplicial complex in \ref{fig:hyper}. The $1$-dimensional complex $ \ini_\prec(\cHH_L) $ represents the minimal $S$-free resolution of the monomial ideal $in_\prec(J_L)$: $$ 0 \longrightarrow S^2 \rightarrowmat{2pt}{4pt}{x_1 & 0 \cr - x_2 & - y_2 \cr 0 & y_3 \cr} S^3 \rightarrowmat{2pt}{4pt}{ x_2 y_3 & x_1 y_3 & x_1 y_2 \cr} S \longrightarrow S/{\ini_\prec(J_L)} \longrightarrow 0 $$ \theorem{initialRes} The minimal free resolution of the initial monomial ideal $\,{\ini}_\prec(J_L) \,$ of the unimodular Lawrence ideal $J_L$ is given by the labeled cell complex ${\ini}_\prec(\cHH_L)$. \proof The $\prec$-positive $1$-dimensional faces $C$ of ${\cHH}^0_L$ are the circuits of $L$, and their labels $m_C$ are the minimal generators $\,{\ini}_\prec(J_L) $. Any higher-dimensional $\prec$-positive face $F$ of ${\cHH}^0_L$ corresponds to a region in the central hyperplane arrangement in $\RR L $ given by the coordinates. Its label $m_F$ is a squarefree monomial by \ref{quotientL}. We can identify $m_F$ with the signed vector in the oriented matroid which represents the region $F$. Now, every vector in an oriented matroid is a conformal union of the circuits [BLSWZ, Proposition 3.7.2]. This implies that $m_F$ is the least common multiple of the labels $m_C$ of all circuits $C$ with $C \subseteq F$. In other words, $m_F$ is the least common multiple of those vertices of $\,{\ini}_\prec(J_L) \,$ which lie on $F \cap H_\prec$. The labeled cell complex ${\ini}_\prec(\cHH_L)$ satisfies the exactness criterion in [BS, Proposition 1.2] because the subcomplex of faces whose label divides $ \xx^\aa \yy^\bb$ is contractible. This follows from a result of Bj\"orner and Ziegler [BLSWZ, Theorem 4.5.7]. Minimality is inherited from $\cHH_L$ since different faces in $\cHH_L$ have different labels. \Box \ref{initialRes} tells us that the minimal cellular resolution $\,(\FF_{\cHH_L/L},\partial)\,$ of a unimodular Lawrence ideal $J_L$ is a {\it universal resolution\/} in the sense that it is stable with respect to any term order $\prec$. This generalizes the fact that the minimal generators of $J_L$ are a {\it universal Gr\"obner basis\/} (\ref{lawrencegens}). In other words, the universal Gr\"obner basis property extends from the generators of $J_L$ to all the higher syzygies. \section{examples} Lawrence ideals arising from graphs \noindent Unimodular lattices arise naturally in the study of directed graphs and matroids. See Chapters 1 and 6 in [Whi1] for a first introduction. Detailed information on unimodularity appears in [Whi2, Chapter 3]. For instance, a famous theorem of Seymour [Whi2, Theorem 3.1.1(9)] states that every unimodular lattice can be built up in a simple way (by duality, $1$-sums, $2$-sums and $3$-sums) from graphic lattices and a certain five-dimensional lattice in $\ZZ^{10}$. In this section we discuss the minimal free resolutions of cographic and graphic lattice ideals in combinatorial terms. Let $G=(V,E)$ be a finite directed graph on $d$ vertices, which we assume are labeled with numbers $1,2,\ldots, d$. Suppose that $E$ has $n$ edges. The edge-node incidence matrix of $G$ is the $n \times d$-matrix whose rows are $\, \ee_i-\ee_j\in \ZZ^d\,$ for $\,(i,j) \in E$. The image of this matrix in $\ZZ^n$ is denoted by $\, L_G$, and is called the {\it graphic lattice\/} of $G$. The orthogonal complement of $L_G$ in $\ZZ^n$ is denoted by $\, L^*_G$, and is called the {\it cographic lattice\/} of $G$. In the language of matroid theory, the graphic lattice $L_G$ is spanned by the {\it cocircuits} of $G$, and the cographic lattice $L^*_G$ is spanned by the {\it circuits\/} of $G$. The following classical result appears in [Whi2, Theorem 1.5.3]. \proposition{graphsareuni} The graphic and cographic lattices $L_G$ and $L^*_G$ are unimodular. We write $J_G = J_{L_G}$ and $J^*_G = J_{L^*_G}$ for the two unimodular Lawrence ideals associated with a directed graph $G$. We call $J_G$ the {\it graphic ideal\/} and $J^*_G$ the {\it cographic ideal\/}. In fact, these ideals depend only on the undirected graph underlying $G$, and they can be described as follows. Replace each edge $(i,j)$ of the directed graph $G = (V,E)$ by two directed edges, one from vertex $i$ to vertex $j$ and the other one from vertex $j$ to vertex $i$. We associate the variables $x_{ij}$ and $x_{ji}$ with these directed edges. This gives a polynomial ring $S$ with $2n$ variables over $k$. We interpret the variable $x_{ji}$ as the homogenizing variable for $x_{ij}$. The pair of variables $\{ x_{ij}, x_{ji} \}$ will play the same role as the pair $\,\{ x_i,y_i \}\,$ in the previous sections. We first discuss the graphic ideal of $G$. It can be computed as an ideal quotient: $$ J_G \quad = \quad \biggl\langle \prod_{j: (i,j) \in E} x_{ij} \,\,\, - \! \prod_{j: (i,j) \in E} x_{ji} \,\,\, \, \big| \,\,\,\, i=1,2,\ldots,d \, \biggr\rangle \,: \, \langle \prod_{(r,s) \in E} x_{rs} x_{sr} \, \rangle^\infty. $$ The $d$ binomials listed above correspond to a lattice basis of $L_G$. Hence $J_G$ has codimension $d$. The minimal generators of $J_G$ are the {\it cocircuits} of the graph $G$. \example{k5} Consider the complete graph on five nodes, $\, G = K_5$. The graphic ideal $J_{K_5}$ is generated by five quartics such as $\, x_{12}x_{13}x_{14}x_{15}-x_{21}x_{31}x_{41}x_{51}\,$ and ten sextics such as $\, x_{13}x_{14}x_{15}x_{23}x_{24}x_{25}-x_{31}x_{32}x_{41}x_{42}x_{51}x_{52} $. They correspond to the $15$ cocircuits of $K_5$. The minimal free resolution of $J_{K_5}$ is given by a $4$-dimensional simplicial complex with $24$ facets, $60$ tetrahedra, $50$ triangles and $15$ edges. \vskip .2cm We generalize this example by describing the minimal free resolution of $J_{K_d}$, the graphic ideal of the complete graph on $d$ nodes. Our construction is related to Lie algebra cohomology; a geometric version was found independently by Bj\"orner and Wachs [BW]. Let $S$ be the polynomial ring over $k$ in the $d(d-1)$ variables $x_{ij}$. Let $F_{r-1}$ denote the free $S$-module whose basis elements correspond to ordered partitions $(A_1\!\mid\! A_2\!\mid\! \ldots\!\mid\! A_r)$ of the set $\set{1,\ldots, d}$, such that $1\in A_1$, for all $1\le r\le d-1$. \theorem{barres} The minimal resolution of the graphic ideal $J_{K_d}$ is the exact complex $$ \FF_\bullet: \quad 0 \rightarrow F_{d-2} \,\, {\buildrel \partial_{d-2} \over \longrightarrow}\,\, F_{d-3} \,\, {\buildrel \partial_{d-3} \over \longrightarrow}\,\, F_{d-4} \,\, \longrightarrow \ldots \longrightarrow \,\, F_2\,\, {\buildrel \partial_2 \over \longrightarrow}\,\, F_1 \,\, {\buildrel \partial_1 \over \longrightarrow}\,\, F_0 \rightarrow 0, \eqno (5.1) $$ where the differential $\partial_{r-1}$ acts on the basis elements of $F_{r-1}$ by the ``cyclic rule'' $$\eqalign{ \partial_{r-1}& (A_1\mid A_2\mid \cdots\mid A_r) \quad = \quad {(-1)}^{r+1}\prod_{i\in A_{r}}\prod_{j\in A_1}x_{ij} \cdot (A_r\cup A_1\mid A_2 \mid \cdots\mid A_{r-1})\cr & + \sum_{s=2}^r{(-1)}^s \! \prod_{i\in A_{s-1}}\prod_{j\in A_s} x_{ij}\cdot (A_1\mid A_2\mid \cdots\mid A_{s-2}\mid A_{s-1}\cup A_s\mid A_{s+1}\mid\cdots\mid A_r). \cr} $$ \proof The central hyperplane arrangement $\cHH_{L_{K_d}}^0$ can be identified with the familiar {\it braid arrangement} which consists of the hyperplanes $z_i = z_j$ in the $(d-1)$-dimensional vector space $\setdef{(z_1,z_2\ldots,z_d) \in \RR^d} {z_1 + \cdots + z_d = 0}$. The $(r-2)$-faces of $\cHH_{L_{K_d}}^0$ are naturally labeled by the ordered partition $(A_1\mid A_2\mid \ldots\mid A_r)$ of $\{1,2,\ldots,d\}$. More precisely, the unique face having a given point in its relative interior is determined by sorting the coordinates of that point. We shall apply the initial ideal construction of \ref{initialRes}. Let $H_\prec$ denote the hyperplane $\{z_1 = 1 \}$ which represents a lexicographic order with the variables $x_{12}, \ldots, x_{1d}$ being highest. The faces of $\cHH_{L_{K_d}}^0$ which have bounded intersection with $H_\prec$ are indexed by those ordered partitions $(A_1\mid A_2\mid \ldots\mid A_r)$ which satisfy $ 1 \in A_1$. The simplicial complex $\ini_\prec(\cHH_{L_{K_d}})$ is the first barycentric subdivision of the $(d-2)$-simplex, as shown for $d=4$ in \nextfig. The vertices of $\ini_\prec(\cHH_{L_{K_d}})$ are indexed by the monomials $\, \prod_{i\in A_1}\prod_{j\in A_2}x_{ij} \,$ which represent partitions $(A_1,A_2)$ with $1 \in A_1$. Higher dimensional faces are labeled by ordered partitions with three or more parts. The cellular resolution given by the labeled simplicial complex $\ini_\prec(\cHH_{L_{K_d}})$ has the format (5.1) with the differential mapping $\, (A_1\mid A_2\mid \cdots\mid A_r) \,$ to $$ \sum_{s=2}^r{(-1)}^s \! \prod_{i\in A_{s-1}}\prod_{j\in A_s} x_{ij}\cdot (A_1\mid A_2\mid \cdots\mid A_{s-2}\mid A_{s-1}\cup A_s\mid A_{s+1}\mid\cdots\mid A_r). $$ This complex is the minimal free resolution of the monomial ideal $\ini_\prec(J_{K_d})$ as in \ref{initialRes}. We lift this complex to the minimal free resolution of $J_{K_d}$ by a construction as in [PS, Theorem 5.4]. Because $\cHH_{L_{K_d}}$ is simplicial, lifting amounts to adding one more term to each differential, and that term is exactly the remaining cyclic term, which is the first one listed in the statement of \ref{barres}. \Box % \capdraw{100}{initial}{ \stext{.5}{1}{\spad{3pt}{\small$x_{13}^{}x_{23}^{}x_{43}^{}$}} % \setext{.245}{.52}{\sepad{3pt}{3pt}{\small$x_{12}^{}x_{13}^{}x_{42}^{}x_{43}^{}$ }} \swtext{.75}{.52}{\swpad{3pt}{3pt}{\small$x_{13}^{}x_{14}^{}x_{23}^{}x_{24}^{}$} } \wtext{.51}{.36}{\wpad{9pt}{\small$x_{12}^{}x_{13}^{}x_{14}^{}$}} % \ntext{0}{.025}{\npad{3pt}{\small$x_{12}^{}x_{32}^{}x_{42}^{}$}} \ntext{.5}{.025}{\npad{3pt}{\small$x_{12}^{}x_{14}^{}x_{32}^{}x_{34}^{}$}} \ntext{1}{.025}{\npad{3pt}{\small$x_{14}^{}x_{24}^{}x_{34}^{} $}} % }{An initial ideal of the complete graphic Lawrence ideal.} % We leave it to a future project to compute the Betti numbers of the graphic ideal $J_{G}$ for arbitrary directed graphs $G$. We expect that there exist nice formulas in terms of the characteristic polynomials of graphic arrangements following [Za]. \vskip .1cm Another open combinatorial problem is to find a formula for the Betti numbers of the {\it cographic ideals} $\, J^*_G $ which can be described as follows. Let $C = (C^+, C^-)$ be a signed circuit of the directed graph $G$; see [BLSWZ, \S 1.1]. This means that $C^+$ and $C^-$ are disjoint subsets of the set of edges such that the edges in $C^+$ together with the reversals of the edges in $C^-$ form a directed cycle which meets each vertex of $G$ at most once. Every signed circuit $C = (C^+,C^-)$ is coded into a binomial: $$ \prod_{(i,j) \in C^+} \prod_{(k,l) \in C^-} x_{ij} x_{lk} \,\quad - \,\, \prod_{(i,j) \in C^+} \prod_{(k,l) \in C^-} x_{ji} x_{kl} .$$ The cographic ideal $J^*_G$ is minimally generated by these binomials where $C = (C^+,C^-)$ runs over all signed circuits of the directed graph $G$. For instance, if $G = K_4$ is the complete graph on $\{1,2,3,4\}$ with edges $(i,j)$ for $i < j$, then there are seven circuits [BLSWZ, bottom of page 3] and our cographic ideal equals $$ \eqalign{ J_{K_4}^* \,\, =\,\, \bigl\langle \, & \, x_{12} x_{23} x_{31} - x_{21} x_{32} x_{13} \,,\,\, x_{12} x_{24} x_{41} - x_{21} x_{42} x_{14} \,,\,\, x_{13} x_{34} x_{41} - x_{31} x_{43} x_{14} ,\cr &x_{23} x_{34} x_{42} - x_{32} x_{43} x_{24} \,,\,\,\,\, x_{12} x_{23} x_{34} x_{41} - x_{21} x_{32} x_{43} x_{14} \,,\,\cr & x_{13} x_{32} x_{24} x_{41} - x_{31} x_{23} x_{42} x_{14} \,,\,\,\,\, x_{12} x_{24} x_{43} x_{31} - x_{21} x_{42} x_{34} x_{13} \,\bigr\rangle \cr} $$ Since the matroid of $K_4$ is self-dual, that is, $K_4$ is self-dual under planar duality of graphs, it follows that the cographic ideal $J_{K_4}^* $ is isomorphic to the graphic ideal $J_{K_4} $ whose minimal resolution is depicted in \ref{fig:initial}. For $d \geq 5$, however, the cographic ideal $J_{K_d}^*$ is different from and more complicated than the graphic ideal $J_{K_d}$. A special case of interest is the complete bipartite graph $K_{d,e}$. Its cographic ideal $J^*_{K_{d,e}}$ is the Lawrence liftings of the ideal of $2 \times 2$-minors of a generic matrix. While the minimal free resolution of the ideals of $2 \times 2$-minors depends on the characteristic of the base field, a result of Hashimoto [RW], their Lawrence liftings have minimal free resolutions which are cellular and characteristic free, by \ref{ZeroOneagain}. \section{diagonal} The diagonal embedding of a unimodular toric variety \noindent We next present a different geometric interpretation of the Lawrence ideal $J_L$. It concerns the diagonal embeddings of a toric variety, written in the homogeneous coordinates of Audin-Cox [Au],[Cox]. This connection first appeared in [Oda]. As an application, Beilinson's spectral sequence is presented for unimodular toric varieties. Let $\Sigma$ be a complete fan in the lattice $\ZZ^m$ and $X$ the associated complete normal toric variety; see e.g.~[Fu]. Let $\bb_1,\dots,\bb_n \in \ZZ^m$ be the primitive generators of the one-dimensional cones of $\Sigma$, and let $B$ be the $n \times m$-matrix with row vectors $\bb_i$. Each $\bb_i$ determines a torus-invariant Weil divisor $D_i$ on $X$, and the group $\Cl(X)$ of torus-invariant Weil divisors modulo linear equivalence has the presentation $$ 0 \,\, \longrightarrow \,\,\ZZ^m \,\,\, {\buildrel B \over \longrightarrow } \,\,\, \ZZ^n \,\,\, {\buildrel \pi \over \longrightarrow } \,\,\, \Cl(X) \,\,\, \longrightarrow 0, $$ % %$$ %0 \, \longrightarrow \, \ZZ^m \, \rightarrowmat{2pt}{4pt}{ B \cr}\, %\ZZ^n \, \rightarrowmat{2pt}{4pt}{ \pi \cr}\, \, \Cl(X) \,\longrightarrow 0, %$$ where $\pi$ takes the $i$-th standard basis vector of $\ZZ^n$ to the linear equivalence class $[D_i]$ of the corresponding divisor $D_i$. If the divisor class group is torsion free, so that $\Cl(X) = \ZZ^{n-m}$, then we express $\pi$ by a matrix $A$, and we are back in (2.1). Audin [Au] and Cox [Cox] define the {\it homogeneous coordinate ring\/} of $X$ to be the polynomial ring $R=k[x_1,\ldots,x_n]$, equipped with a grading by the abelian group $\Cl(X)$ via the morphism $\pi$ above. Hence a monomial $\xx^\aa\in R$ has degree $[\sum_{i=1}^n a_i D_i]\in\Cl(X)$. The fan $\Sigma$ is encoded in the {\it irrelevant ideal} $B_\Sigma \subset R$ whose monomial generators correspond to complements of facets of $\Sigma$. Coherent sheaves on the toric variety $X$ are represented by $B_\Sigma$-torsion-free $\Cl(X)$-graded modules over $R$; see [Cox] for the case where $\Sigma$ is simplicial, and [Mu] for the general case. Closed subschemes of $X$ are defined by $B_\Sigma$-saturated $\Cl(X)$-graded ideals of $R$. Furthermore, for any $\Cl(X)$-graded $R$-module $T$, and any $\aa\in \Cl(X)$, we define its {\it twist} $T(\aa)$ to be the graded $R$-module with components $T(\aa)_\bb:=T_{\aa+\bb}$. Let $\cOO_X(\aa)$ denote the coherent sheaf on $X$ corresponding to the twisted $R$-module $R(\aa)$. The toric variety $X \times X$ has the homogeneous coordinate ring $$ S \quad = \quad R \, \otimes_k \, R \quad = \quad k[x_1,\ldots,x_n, \, y_1,\ldots,y_n] $$ The diagonal embedding $X\subset X\times X$ defines a closed subscheme, and is represented by a $\Cl(X)\times\Cl(X)$-graded ideal $I_X$ in $S$. That ideal $I_X $ is the kernel of the natural map $\,\, S = R \otimes R \, \rightarrow \, k[\Cl(X)]\otimes R \, ,\,\, \xx^\uu \yy^\vv = \xx^\uu \otimes \xx^\vv \, \mapsto \, [\uu]\otimes\xx^{\uu+\vv}$. Explicitly, $$I_X \quad = \quad \idealdef{\xx^\uu \yy^\vv - \xx^\vv \yy^\uu} {\pi(\uu)=\pi(\vv)\, \, \hbox{in} \,\, \Cl(X)} \quad \subset \quad S .$$ This formula proves the following observation. \proposition{LawrenceGivesDiagonal} The ideal $I_X \subset S $ defining the diagonal embedding $X \subset X \times X$ equals the Lawrence ideal $J_L$ for the lattice $L=\im(B)=\ker(\pi)$ of principal divisors. The basic example is projective space $X = \PP^m$. Here $n= m+1$, $\, L = \ker (\, 1 \, \,1 \, \,1 \, \, \cdots \, 1 \,) $, $\Cl(X) = \ZZ^{1}$, and $I_X = J_L$ is the ideal (1.1) of $2 \times 2$-minors of a $\, 2 \times n$-matrix of indeterminates. The minimal free resolution of $I_X = J_L$ is an Eagon-Northcott complex. Example 10 in [Oda] demonstrates how Beilinson's spectral sequence for $\PP^m$ can be derived from this observation. The results in this paper provide a cellular model for this Eagon-Northcott complex as a hypersimplicial complex (Figure 1). We will show that this derivation of Beilinson's spectral sequence extends to the general setting of \ref{unimod-toric-syz}. Thus our cellular resolutions provide a partial solution to [Oda, Problem 6]. We say that a toric variety $X$ is {\it unimodular} if its lattice $L$ of principal divisors is a unimodular sublattice of $\ZZ^n$. This condition is equivalent, by [Stu, Remark 8.10], to saying that the toric variety $X$ is smooth and any other variety obtained by toric flips and flops is smooth as well. Toric varieties with this property appear frequently in representation theory and in integer programming. For example, the toric varieties associated with {\it transportation polytopes} and {\it products of minors} are unimodular. These toric varieties were studied recently by Babson and Billera [BB]. Suppose now that $X$ is a unimodular toric variety. Then \ref{unimod-toric-syz} constructs an $m$-dimensional cellular model for the $\Cl(X)\times\Cl(X)$-graded minimal resolution for the ideal $\, I_X \subset S \,$ of the diagonal embedding of $X$ in $X\times X$: $$L_\bullet: \quad 0\longrightarrow L_m \longrightarrow \ldots \longrightarrow L_2\longrightarrow L_1 \longrightarrow L_0=\cOO_{X\times X} \longrightarrow \cOO_{X\times X}/\cII_X \longrightarrow 0. $$ Here we identify $I_X$ with the associated coherent sheaf $\cII_X \subset\cOO_{X\times X}$, which is the ideal sheaf of the diagonal embedding of $X$. Note that $S/I_X$ is Cohen-Macaulay. Since $\cII_X$ is a $\Cl(X)\times\Cl(X)$-homogeneous ideal, we can write $\, L_i \, = \,\oplus_{j=1}^{m_i}\cOO(-\aa_{ij},-\bb_{ij})\,$ with $\aa_{ij}, \bb_{ij}\in\Cl(X)$. Here $\cOO(-\aa_{ij},-\bb_{ij})=\cOO(-\aa_{ij})\boxtimes\cOO(-\bb_{ij})= p_1^*(\cOO(-\aa_{ij}))\otimes p_2^*(\cOO(-\bb_{ij}))$ denotes the exterior tensor product on $X\times X$ of $\cOO(-\aa_{ij})$ and $\cOO(-\bb_{ij})$, and $p_i$ are the projections on the factors of $X\times X$. Let $\cFF$ be any coherent sheaf on $X$. By tensoring the above free resolution $L_\bullet$ with $p_1^*(\cFF)$ we obtain a resolution $C_\bullet$ for the restriction of $p_1^*(\cFF)$ to the diagonal $X \subset X\times X$. As in the case of projective space [Bei], pushing via ${p_2}_*$ a Cartan-Eilenberg injective resolution of $C_\bullet$ yields a double complex of $\cOO_X$-modules, whose total chain complex has as cohomology the hyperdirect image ${\mathbf R}{p_2}_*(C_\bullet)$. One of the spectral sequences belonging to the double complex ${\mathbf R}{p_2}_*(C_\bullet)$ yields the following version of the {\it Beilinson Spectral Sequence}. It says that any coherent sheaf $\cFF$ can be reconstructed from the knowledge of the cohomology of a few of its $\Cl(X)$-twists. \theorem{beilinson.ss} Let $\cFF$ be a coherent sheaf on a complete unimodular toric variety $X$. Then there is a (third quadrant) spectral sequence with $E_1$ term $$E_1^{pq} \quad = \quad \bigoplus_j H^q(\cFF\otimes\cOO_X(-\aa_{pj}))\otimes\cOO_X(-\bb_{pj})$$ which converges to the associated graded sheaf of a filtration of $\cFF$. \Box In the case of $\PP^m$, Beilinson's spectral sequence induces an equivalence between the derived category $\DD^b(\Coh(\PP^m))$ of bounded complexes of sheaves with coherent cohomology, and the derived category of modules over the exterior algebra in $m+1$ generators. An analogous simple description of $\DD^b(\Coh(X))$ does not hold over an arbitrary unimodular toric variety $X$, since the bundle $\cEE=\oplus_{i,j}\cOO(-\aa_{ij})$ defined as the direct sum (without repetitions) of all ``left'' summands in the above resolution of the ideal sheaf of the diagonal is in general not exceptional, i.e. $\Ext^i_{\cOO_X}(\cEE,\cEE)\ne 0$ for some $i>0$. See [AH] for a related approach to describing $\DD^b(\Coh(X))$. %=========================================================================% \references %=========================================================================% \itemitem{[AH]} K.~Altmann, L.~Hille: Strong exceptional sequences provided by quivers. {\sl Algebr. Represent. Theory} {\bf 2} (1999) 1-17. \itemitem{[Au]} M.~Audin: {\sl The Topology of Torus Actions on Symplectic Manifolds}, Progress in Math. {\bf 93}, Birkh\"auser, 1991. \itemitem{[BB]} E.~Babson, L.J.~Billera: The geometry of products of minors, {\sl Discrete Comput. Geometry} {\bf 20} (1998) 231--249 \itemitem{[BPS]} D.~Bayer, I.~Peeva, B.~Sturmfels: Monomial resolutions, {\sl Math. Res. Lett.} {\bf 5} (1998), no. 1-2, 31--46. \itemitem{[BS]} D.~Bayer, B.~Sturmfels: Cellular resolutions of monomial modules, {\sl J. Reine Angew. Math.} {\bf 502} (1998), 123--140. \itemitem{[Bei]} A.~Beilinson: Coherent sheaves on $\PP^N$ and problems of linear algebra, {\sl Funct. Anal. Appl.}, {\bf 12}, (1978), 214--216. \itemitem{[BLSWZ]} A.~Bj\"orner, M.~Las~Vergnas, B.~Sturmfels, N.~White and G.~Ziegler: {\sl Oriented Matroids}, Cambridge University Press, 1993. \itemitem{[BW]} A.~Bj\"orner, M.~Wachs: Geometrically constructed bases for intersection lattices of real arrangements, preprint 1999. \itemitem{[Cox]} D.A.~Cox: The homogeneous coordinate ring of a toric variety, {\sl J. Algebraic Geom.} {\bf 4} (1995), no. 1, 17--50. \itemitem{[Fu]} W.~Fulton: {\sl Introduction to Toric Varieties}, Princeton Univ.~Press, 1993. %\itemitem{[Has]} M.~Hashimoto: %Determinantal ideals without minimal free resolutions, %{\sl Nagoya Math. J.} {\bf 118} (1990) 203--216. \itemitem{[Mu]} M.~Musta\c t\u a: Vanishing theorems on toric varieties, preprint 1999. \itemitem{[Oda]} T.~Oda: Problems on Minkowski sums of convex lattice polytopes, Manuscript, 1997. \itemitem{[PS]} I.~Peeva, B.~Sturmfels: Generic lattice ideals, {\sl J. Amer. Math. Soc.}, {\bf 11}, (1998), no. 2, 363--373. \itemitem{[RW]} J.~Roberts, J.~Weyman: A short proof of a theorem of M. Hashimoto. {\sl J. Algebra} {\bf 134} (1990), no.~1, 144--156. \itemitem{[Stu]} B.~Sturmfels: {\sl Gr\"obner Bases and Convex Polytopes}, AMS University Lecture Series, Vol. {\bf 8}, Providence RI, 1995. \itemitem{[Whi1]} N.~White: {\sl Theory of Matroids}, Encyclopedia of Mathematics and its Applications, {\bf 26}, Cambridge University Press, 1986. \itemitem{[Whi2]} N.~White: {\sl Combinatorial Geometries}, Encyclopedia of Mathematics and its Applications, {\bf 29}, Cambridge University Press, 1987. \itemitem{[Za]} T.~Zaslavsky: Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, {\sl Mem. Amer. Math. Soc.} {\bf 1} (1975), no.~154. \itemitem{[Zi]} G.~Ziegler: {\sl Lectures on Polytopes}, Graduate Text in Mathematics {\bf 152}, Springer Verlag, 1997. \vskip .6cm \widow{.1} \noindent Dave Bayer, Department of Mathematics, Barnard College, Columbia University, New York, NY 10027, USA, {\tt bayer@math.columbia.edu}. \vskip .2cm \noindent Sorin Popescu, Department of Mathematics, Columbia University, New York, NY 10027, USA, and Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794, {\tt psorin@math.columbia.edu}. \vskip .2cm \noindent Bernd Sturmfels, Department of Mathematics, University of California, Berkeley, CA 94720, USA, {\tt bernd@math.berkeley.edu}. \bye