%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % The moduli space of $(1,11)$-polarized abelian surfaces is % unirational % % by Mark Gross & Sorin Popescu % 02/01/98 % ( Plain TeX with diagrams.tex) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \font\hd=cmbx10 scaled\magstep1 \magnification=\magstep 1 \input diagrams.tex % defs2.tex %================================================ % F o n t s %================================================ % define \Cal as control sequence with an argument for \fam2 in math mode \def\Cal#1{{\fam2#1}} %------------------------------------------------ % For the following definitions of font families and % corresponding control sequences see: % AMSFonts 2.0 User's Guide page 12. % There must no other families be defined before these, % if you want to refer to the \fam numbers % of the families starting from number 8 for \msamfam. %------------------------------------------------ % define \Msa as control sequence with an argument for msam in math mode: \font\tenmsam=msam10 \font\sevenmsam=msam7 \font\fivemsam=msam5 \newfam\msamfam % family "8 (if no other \newfam before) \textfont\msamfam\tenmsam \scriptfont\msamfam\sevenmsam \scriptscriptfont\msamfam\fivemsam \def\Msa#1{{\fam\msamfam\relax#1}} % % define \MSAM as control sequence to switch to the msam font. \def\MSAM{\fam\msamfam\tenmsam} %------------------------------------------------ % define \Bbb#1 as control sequence with an argument for msbm in math mode: \font\tenmsbm=msbm10 \font\sevenmsbm=msbm7 \font\fivemsbm=msbm5 \newfam\msbmfam % family "9 (if no other \newfam before) \textfont\msbmfam\tenmsbm \scriptfont\msbmfam\sevenmsbm \scriptscriptfont\msbmfam\fivemsbm \def\Bbb#1{{\fam\msbmfam\relax#1}} % % define \MSBM as control sequence to switch to the msbm font. \def\MSBM{\fam\msbmfam\tenmsbm} %------------------------------------------------ % define \frak#1 as control sequence with an argument for eufm in math mode: \font\teneufm=eufm10 \font\seveneufm=eufm7 \font\fiveeufm=eufm5 \newfam\eufmfam % family "A (if no other \newfam before) \textfont\eufmfam\teneufm \scriptfont\eufmfam\seveneufm \scriptscriptfont\eufmfam\fiveeufm \def\frak#1{{\fam\eufmfam\relax#1}} % % define \Fraktur as control sequence to switch to the eufm font. \def\Fraktur{\fam\eufmfam\teneufm} %------------------------------------------------ % %=========================================================== % D e f i n i n g N e w S y m b o l s %=========================================================== % Definition of new symbols from the AMS-Fonts % by the AMS-identification number (in hex): % % The first digit is the AMS family number: % 1 for the MSAM family and 2 for the MSBM family. % % The second digit is the class number (for % example 2 for a binary operator, or 0 for % an ordinary character). Same as in Plain TEX, but in Plain TEX % this is the first digit, and the second is the family number. % % The last 2 digits give the position in the font table. % % Therefore the macro must replace the first digit by the corresponding % Plain TEX font family number and then interchange the first two digits % in order to switch to the Plain Tex identification (ID-) number. % (See Schwarz, Einf?hrung in TEX page 256 \mathchar (or The TEX Book % p. 154) and page 264 \newfam. % %================================================ % % 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 %======================================================================== % The following macro " \newmathsymbol " takes as % #1 the name of the new control sequence, % #2 the Plain TEX number of the font family, a decimal number 0-{15} or % in the invariant form \msamfam, \msbmfam ,\eufmfam, ... which is % independent of the sequence in which those families are loaded % #3 the Plain TEX class number as hex digit 0-7 % #4#5 the position number of the character in the font table as a % two digit hex number 00-7F. %------------------------------------------------------------------------ \newcount\famcnt % define new registers to \newcount\classcnt % store the digits for conversion % \newcount\positioncnt \newcount\codecnt %------------------------- \def\newmathsymbol#1#2#3#4#5{ % #1 is the name of the new % % macro to be defined \famcnt=#2 % #2 is the font family number in decimal \multiply\famcnt by "100 % make it the third hex digit \classcnt="#3 % #3 is the class number (same as in Plain) \multiply\classcnt by "1000 % make it the fourth hex digit \positioncnt="#4#5 % #4#5 is the hex position in the fonttable % \codecnt=\classcnt % add the three numbers to obtain \advance\codecnt by \famcnt % the Plain TEX identification number \advance\codecnt by \positioncnt % \mathchardef#1=\codecnt} % define #1 to become a character % % with this identification number %======================================================================= \def\span{{\rm span}_{\bf C}} %rational short map \def\rto{\raise.5ex\hbox{$\scriptscriptstyle ---\!\!\!>$}} \def\Pseven{{\bf P^7}} \def\Psix{{\bf P^6}} \def\Pfive{{\bf P^5}} \def\Pfour{{\bf P^4}} \def\Pthree{{\bf P^3}} \def\Pthreedual{\check\Pthree} \def\Ptwo{{\bf P^2}} \def\Pone{{\bf P^1}} \def\P{{\bf P}} \def\R{{\bf R}} \def\genus#1{p_a(#1)} \def\half{{1 \over 2}} \def\mun#1{\mu_n^{\otimes #1}} \def\qlzl{{\bf Q}_l/{\bf Z}_l} \def\A{{\cal A}} \def\C{{\cal C}} \def\D{{\cal D}} \def\SS{{\cal S}} \def\X{{\cal X}} \def\O{{\cal O}} \def\L{{\cal L}} \def\PP{{\cal P}} \def\T{{\cal T}} \def\H{{\cal H}} \def\G{{\cal G}} \def\V{{\cal V}} \def\F{{\cal F}} \def\I{{\cal I}} \def\M{{\cal M}} \def\N{{\cal N}} \def\E{{\cal E}} \def\K{{\cal K}} \def\ZZ{{\cal Z}} \def\HH{{\underline H}} \def\HHH{{\bf H}} \def\En{{\cal EN}} \def\EY{\E(1)|_Y} \def\EYp{\E'(1)|_Y} \def\PEY{\P(\EY)} \def\PEYp{\P(\EYp)} \def\OP{\O_{\PEY}(1)} \def\OP1{\O_{\Pone}} \def\OPp{\O_{\PEYp}(1)} \def\OPtt{\Omega_{\Pthree}(2)} \def\dpdp{\pi^*\Omega_{\Pfour}} \def\Pic{{\rm Pic}} \def\Spec{\mathop{\rm Spec}} \def\coker{\mathop{\rm coker}} \def\Im{\mathop{\rm Im}} \def\im{\mathop{\rm im}} \def\Brp{\mathop{\rm Br'}} \def\Br{\mathop{\rm Br}} \def\char{\mathop{\rm char}} \def\Gr{\mathop{\rm Gr}\nolimits} \def\GR{\Gr(1,\Pthree)} \def\grdn{\Gr(d,\P^n)} \def\Proj{\mathop{\rm Proj}} \def\Sym{\mathop{\rm Sym}\nolimits} \def\mod{\mathop{\rm mod}} \def\ext{\mathop{\rm Ext}\nolimits} \def\hom{\mathop{\rm Hom}\nolimits} \def\supp{\mathop{\rm Supp}} \def\rank{\mathop{\rm rank}\nolimits} \def\depth{\mathop{\rm depth}} \def\homd{\mathop{\rm hd}} \def\chom{{\cal HOM}} \def\cext{{\cal EXT}} \def\lhom{{\underline{\hom}}} \def\lext{{\underline{\ext}}} \def\boldz{{\bf Z}} \def\boldp{{\bf P}} \def\hens{{\O_{S,\bar s}}} \def\Gm{{\bf G}_m} \def\Gmm#1{{\bf G}_{m,#1}} \def\dual#1{{#1}^{\scriptscriptstyle \vee}} \newsymbol\SEMI226F \def\rtimes{\mathop{\SEMI}} \def\qz{{\bf Q}/{\bf Z}} \def\qzl{{\bf Q}_l/{\bf Z}_l} \def\exact#1#2#3{0\rightarrow#1\rightarrow#2\rightarrow#3\rightarrow0} \def\cohomology#1#2#3{\eqalign{ 0\rightarrow H^0(#1)\rightarrow H^0(#2) &\rightarrow H^0(#3) \rightarrow H^1(#1)\cr &\rightarrow H^1(#2) \rightarrow H^1(#3) \rightarrow 0.\cr}} \def\norm{\N_{Y/Q}} \def\Ku{K^{\cdot}} \def\Kl{K_{\cdot}} \def\clu{cl^{\cdot}} \def\cll{cl_{\cdot}} \def\adj{\phi_{\O(H)}} \def\fdf{\left\lfloor{d\over 4}\right\rfloor} \def\cdc{\left\lceil{d\over 4}\right\rceil} %\def\normalbaselines{\baselineskip20pt %\lineskip3pt \lineskiplimit3pt } \def\mapleft#1{\smash{ \mathop{\longleftarrow}\limits^{#1}}} \def\mapright#1{\smash{ \mathop{\longrightarrow}\limits^{#1}}} \def\mapsw#1{\smash{ \mathop{\swarrow}\limits^{#1}}} \def\mapse#1{\smash{ \mathop{\searrow}\limits^{#1}}} \def\mapdown#1{\Big\downarrow \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} \def\ee{\eta\times\eta} \def\eep{\eta\times\eta'} \def\epe{\eta'\times\eta} \def\epep{\eta'\times\eta'} \def\yty{Y\times Y} \def\ytyt{\tilde{\yty}} \def\qtq{Q\times Q} \def\qtqt{\tilde{\qtq}} \def\ss{\sigma^*} \def\sh{\check s} \def\pts{p_2^*} % defs3.tex %================================================ % M a t h. O p e r a t o r s %================================================ % \def\Hom{\mathop{\rm Hom}} % \def\Ext{\mathop{\rm Ext}} % \def\Tor{\mathop{\rm Tor}} % \def\Char{\mathop{\rm char}} % \def\End{\mathop{\rm End}} % \def\ker{\mathop{\rm ker}} % \def\coker{\mathop{\rm coker}} % \def\im{\mathop{\rm im}} % \def\gr{\mathop{\rm gr}} % \def\Quot{\mathop{\rm Quot}} % \def\dsum{\mathop{\oplus}} % \def\bdsum{\mathop{\bigoplus}} % \def\rk{\mathop{\rm rk}} % \def\span{{\rm span}_{\bf C}} % \def\deg{\mathop{\rm deg}} % \def\dim{\mathop{\rm dim}} % \def\Wedge{\mathop{\Lambda}} % %========================================================== % O t h e r C o n t r o l S e q u e n c e s %========================================================== % qed in textmode \newsymbol\QED1003 \def\qed{\hbox{$\QED$}} %---------------------------------------------------------- % semi-direct product \newsymbol\SEMI226F \def\rtimes{\mathop{\SEMI}} %------------------------------------------------------------ % define font "line10" as "lineten" and generate a control % sequence "arrow" which prints the wanted character from %line10 by its octal number: \font\lineten=line10 \def\arrow#1{\lineten\char'#1} %------------------------------------------------------------ % bold >= from AMS-TeX \font\boldmath=cmbsy10 \def\bbiger{\boldmath\char'025} %------------------------------------------------------------ % rightarrow with symbols above: #1 symbols on arrow: \def\mapright#1{{\smash{\mathop{\longrightarrow}\limits^{#1}}}} %------------------------------------------------ % leftarrow with symbols above: \def\mapleft#1{\smash{\mathop{\longleftarrow}\limits^{#1}}} %---------------------------------------------------------------- % downarrows with symbols : \def\mapdown#1{\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} \def\bmapdown#1{\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} %------------------------------------------------------------------- % upwards arrows with symbols : \def\mapup#1{\uparrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} \def\bmapup#1{\Big\uparrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} %-------------------------------------------------------------- % hooked rightarrow with symbols : \def\hookmapright#1{\smash{\mathop{\hookrightarrow}\limits^{#1}}} %-------------------------------------------------------------------- % long rightarrow with symbols above: % #1: symbols on arrow, #2: length of the extension in cm. \def\harrowext#1{\hskip-3.5pt\raise2.3pt\hbox to#1{\hrulefill}\hskip-3.5pt} % \def\longmapright#1#2{\hbox{${\mathop{\harrowext{#2cm}% \longrightarrow}\limits^{#1}}$}} %--------------------------------------------------------------- %long left with symbols above: \def\longmapleft#1#2{\hbox{${\mathop{% \longleftarrow}\limits^{#1}\harrowext{#2cm}}$}} %---------------------------------------------------------------- % nwarrow with symbols above: \def\snwarrow#1{\smash{\mathop{\nwarrow}\limits_{#1}}} %---------------------------------------------------------------- % nwarrow with symbols above: \def\snwarrowb#1{\smash{\mathop{\nwarrow}\limits^{#1}}} %---------------------------------------------------------------- % long north west arrow % \def\longnwarrow{\hskip-15pt\lower6pt\hbox{\vbox{\hbox{$\nwarrow$} \vskip-1.5pt\hbox{\hskip7.95pt\arrow100 }}}} %---------------------------------------------------------------- %rational short map \def\rto{\raise.5ex\hbox{$\scriptscriptstyle ---\!\!\!>$}} %----------------------------------------------------------------- % New line in hmode: \def\newln{\hfil\break} % % newline more general \def\nline{\par\noindent} % % uparrow two characters long \def\twoup{\vbox{\hbox{\hskip-2pt$\uparrow$} \vskip-3pt\hbox{\hskip-1pt$|$}}} % % uparrow three characters long \def\threeup{\vbox{\hbox{\hskip-2pt$\uparrow$} \vskip-3pt\hbox{\hskip-1pt$|$} \vskip-3pt\hbox{\hskip-1pt$|$}}} % % shortmid from AMS-TeX \newsymbol\shortmid2370 %========================================================== % \def\twitle{\sim} %============================================================ % S h e a v e s %============================================================ % ideal sheaf: \def\id#1#2{{\cal I}_{#1}(#2)} % % twisted structural sheaf: \def\struct#1#2{{\cal O}_{#1}(#2)} % % structural sheaf: \def\sstruct#1{{\cal O}_{#1}} % % twisted structural sheaf (simplified) : \def\systruct#1{{\cal O}(#1)} % % sheaf of differentials (simplified) : \def\somega#1#2{\Omega^{#1}(#2)} % %%%%%%%%%%%%%%%%%%%% \def\Pzz{\Ptwo^-/\boldz_2\times\boldz_2} \def\Ptzz{\Pthree^-/\boldz_2\times\boldz_2} \def\CC{{\bf C}} \def\QQ{{\bf Q}} \def\RR{{\bf R}} \def\NN{{\bf N}} \def\vt{\vartheta} \def\dim{{\rm dim}\,} \def\deg{{\rm deg}\,} \def\Pf{{\rm Pf}\,} % Macros from defs3.tex \def \PSL {\mathop{\rm PSL}\nolimits} \def \SL {\mathop{\rm SL}\nolimits} \def \GL {\mathop{\rm GL}\nolimits} \def\PGL {\mathop{\rm PGL}\nolimits} \def\vt{\vartheta} \def\dim{\mathop{\rm dim}\nolimits} \def\deg{\mathop{\rm deg}\nolimits} \def\Pf{{\rm Pf}\,} \def \Sec {\mathop{\rm Sec}\nolimits} \font\smc=cmcsc10 \font\abst=cmr9 \font\smallsmc=cmcsc8 \font\smallrm=cmcsc8 \headline={\ifodd\pageno \ifnum\pageno>1 \smallrm \hfil The moduli space of $(1,11)$-polarized abelian surfaces is unirational % running head for right-hand page is title in caps \hfil\folio \else\hfill\fi \else \smallrm \folio \hfill M.~Gross and S.~Popescu % running head for left-hand page is authors in caps \hfill\fi} \footline={\hss} % footline is blank % begin.tex %=========================================================================% % load begin.tex only once, but keep count to match \bye commands %=========================================================================% \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\eject\nopagenumbers\par\vfill\supereject\end} \else\def\endcmd{}\fi \endcmd } %=========================================================================% % initialize TeX %=========================================================================% \baselineskip=15pt %\hsize = 6truein %\hoffset = 0.5truein %\vsize = 8.5truein \voffset = 0.2truein \emergencystretch = 0.05\hsize \parskip=3pt plus1pt minus.5pt \overfullrule=0pt \newif\ifblackboardbold % comment out the following line if AMS msbm fonts aren't available \blackboardboldtrue %=========================================================================% % select fonts %=========================================================================% \font\titlefont=cmbx12 scaled\magstep1 \font\sectionfont=cmbx10 scaled\magstephalf \font\netfont=cmtt9 % 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 %=========================================================================% % font size-changing command ("A Beginner's Book of TeX" p35, p275) %=========================================================================% \font\Arm=cmr8 \font\Ai=cmmi8 \font\Asy=cmsy8 \font\Abf=cmbx8 \font\Brm=cmr6 \font\Bi=cmmi6 \font\Bsy=cmsy6 \font\Bbf=cmbx6 \font\Crm=cmr5 \font\Ci=cmmi5 \font\Csy=cmsy5 \font\Cbf=cmbx5 \ifblackboardbold \font\Abbold=msbm10 at 8pt \font\Bbbold=msbm7 at 6pt \font\Cbbold=msbm5 \fi \def\smallmath{% \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 } %=========================================================================% % 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{\it 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 \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 \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} \othernumbered{claim}{Claim} \othernumbered{question}{Question} %=========================================================================% % 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. %\input epsf \newcount\figcount \figcount=0 \newbox\drawing \newcount\drawbp \newdimen\drawx \newdimen\drawy \newdimen\ngap \newdimen\sgap \newdimen\wgap \newdimen\egap \def\drawbox#1#2#3{\vbox{ \setbox\drawing=\vbox{\offinterlineskip\epsfbox{#2.eps}\kern 0pt} \drawbp=\epsfurx \advance\drawbp by-\epsfllx\relax \multiply\drawbp by #1 \divide\drawbp by 100 \drawx=\drawbp truebp \ifdim\drawx>\hsize\drawx=\hsize\fi \epsfxsize=\drawx \setbox\drawing=\vbox{\offinterlineskip\epsfbox{#2.eps}\kern 0pt} \drawx=\wd\drawing \drawy=\ht\drawing \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} \advance\figcount by 1 \goodbreak \midinsert \centerline{\ifgridlines\boxgrid\drawing\fi\box\drawing} \smallskip \vbox{\offinterlineskip \centerline{Figure~\number\figcount} \smash{\marginlabel{#2}}} \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=\hbox{#5} \dp\figtbox=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}\dp0=0pt\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}}} % 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}} %=========================================================================% % macros for matrices and arrows %=========================================================================% % typical usage: % \rightarrowmat{2pt}{4pt}{d & bd \cr \!-c & 0 \cr 0 & -ac \cr} \def\bmatrix#1#2{{\smallmath\left[\vcenter{\halign {&\kern#1\hfil$##\mathstrut$\kern#1\cr#2}}\right]}} \def\rightarrowmat#1#2#3{ \setbox1=\hbox{\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{\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{\smallmath #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{\smallmath #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. % \preview squeezes two pages side by side in landscape orientation. It % is not suitable for printing, but ideal for previewing on a two page % monitor. % \twoup squeezes two pages onto letter paper in landscape mode, % suitable for printing. % Each of these macros calls the file "quire.tex" \def\bookletdims{ \hsize=5.25truein \vsize=7truein } \def\legalbooklet#1{ \input quire \bookletdims \htotal=7.0truein \vtotal=8.5truein % below computed from above \hoffset=\htotal \advance\hoffset by -\hsize \divide\hoffset by 2 \voffset=\vtotal \advance\voffset by -\vsize \divide\voffset by 2 \advance\voffset by -.0625truein \shhtotal=2\htotal % below doesn't need to change \horigin=0.0truein \vorigin=0.0truein \shstaplewidth=0.01pt \shstaplelength=0.66truein \shthickness=0pt \shoutline=0pt \shcrop=0pt \shvoffset=-1.0truein \ifnum#1>0\quire{#1}\else\qtwopages\fi } \def\preview{ \input quire \bookletdims \hoffset=0.1truein \vtotal=8.5truein \shhtotal=14truein % below computed from above \voffset=\vtotal \advance\voffset by -\vsize \divide\voffset by 2 \advance\voffset by -.0625truein \htotal=2\hoffset \advance\htotal by \hsize % below doesn't need to change \horigin=0.0truein \vorigin=0.0truein \shstaplewidth=0.5pt \shstaplelength=0.5\vtotal \shthickness=0pt \shoutline=0pt \shcrop=0pt \shvoffset=-1.0truein \qtwopages } \def\twoup{ \input quire \hsize=4.79452truein % 5.25/1.095 \vsize=7truein \vtotal=8.5truein \shhtotal=11truein % below computed from above \hoffset=-2\hsize \advance\hoffset by \shhtotal \divide\hoffset by 6 \voffset=\vtotal \advance\voffset by -\vsize \divide\voffset by 2 \advance\voffset by -12truept \htotal=2\hoffset \advance\htotal by \hsize % below doesn't need to change \horigin=0.0truein \vorigin=0.0truein \shstaplewidth=0.01pt \shstaplelength=0pt \shthickness=0pt \shoutline=0pt \shcrop=0pt \shvoffset=-1.0truein \qtwopages } %=========================================================================% % 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} %========================================================================== % macros (specific to this paper) %========================================================================== % 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\NN{\enma{\mathbb{N}}} \def\ZZ{\enma{\mathbb{Z}}} % caligraphic symbols \def\Ch{\enma{\cal C}} % bold symbols \def\aa{\enma{\mathbf{a}}} \def\bb{\enma{\mathbf{b}}} \def\cc{\enma{\mathbf{c}}} \def\xx{\enma{\mathbf{x}}} \def\mm{\enma{\mathbf{m}}} % misc \def\th{{^{\rm th}}} \def\H{\widetilde{H}} \def\mtext#1{\;\,\allowbreak\hbox{#1}\allowbreak\;\,} \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\depth{\mathop{\rm depth}\nolimits} \def\reg{\mathop{\rm reg}\nolimits} \def\Ext{\mathop{\rm Ext}\nolimits} \def\Tor{\mathop{\rm Tor}\nolimits} \def\link{\mathop{\rm lk}\nolimits} \def\star{\mathop{\rm star}\nolimits} \def\core{\mathop{\rm core}\nolimits} \def\abs#1{\enma{\left| #1 \right|}} \def\GCD{\mathop{\rm GCD}\nolimits} \def\In{\mathop{\rm In}\nolimits} \def\Gin{\mathop{\rm Gin}\nolimits} \def\Ginc{\mathop{\rm Ginc}\nolimits} \def\red{\mathop{\rm red}\nolimits} \def\Ker{\mathop{\rm Ker}\nolimits} \def\Hom{\mathop{\rm Hom}\nolimits} \def\Syz{\mathop{\rm Syz}\nolimits} \centerline{\hd The moduli space of (1,11)-polarized abelian surfaces is unirational} \medskip \centerline{\it Mark Gross\footnote{*}{Supported by NSF grant DMS-9700761.} and Sorin Popescu\footnote{**} {Partially supported by NSF grant DMS-9610205 and MSRI, Berkeley.}} \medskip %\centerline{\timestamp} \medskip {\settabs 3 \columns \+Mathematics Institute&& Department of Mathematics\cr \+University of Warwick&& Columbia University\cr \+Coventry, CV4 7AL, UK&& New York, NY 10027\cr \+mgross@maths.warwick.ac.uk&& psorin@math.columbia.edu\cr} {\vglue .5in \narrower{\noindent \abst Abstract. We prove that the moduli space $\A_{11}^{lev}$ of $(1,11)$-polarized abelian surfaces with level structure of canonical type is birational to Klein's cubic hypersurface in $\Pfour$. Therefore, $\A_{11}^{lev}$ is unirational but not rational, and there are no $\Gamma_{11}$-cusp forms of weight 3. The same methods also provide an easy proof of the rationality of $\A_{9}^{lev}$.} \vglue .5in} %\showlabelsabove % this prints the labels above each def, thm, etc %\showlabels % this prints the labels on the side %\preview % preview format 2 pages side by side % Forward references \forward{torus}{Lemma}{3.4} \forward{lin.eq}{Theorem}{2.6} Classical results of Tai, Freitag and Mumford and newer results of O'Grady, Gritsenko, Hulek and Sankaran say that moduli spaces of polarized abelian varieties are almost always of general type. However, for abelian varieties of small dimension and polarizations of small degree the situation is different and the corresponding moduli spaces usually have beautiful geometry. In this paper we describe a projective model for the moduli of complex abelian surfaces with a polarization of type $(1,11)$, with level structure of canonical type. As a direct consequence we obtain the unirationality of this moduli space, which also turns out to be non-rational. However, unirationality already implies that there exist no $\Gamma_{11}$-cusp forms of weight 3. Let $\A_d$ denote the moduli space of polarized abelian surfaces of type $(1,d)$, and let $\A_d^{lev}$ be the moduli space of $(1,d)$-polarized abelian surfaces with canonical level structure. The map which forgets the level structure represents $\A_d^{lev}$ as a finite cover of $\A_d$. Its general fiber is $\Gamma_d/\Gamma_d^{lev}\cong \SL_2(\boldz_{d})$, where $\Gamma_d$ and $\Gamma_d^{lev}$ denote the corresponding paramodular groups. In particular, if $d$ is an odd prime number, then the forgetful morphism is a ramified cover of degree $d(d^2-1)/2$. (See [LB], [Mum], [GP1] for the definition of canonical level structure, and basic results.) Our main result is the following \theorem{main} The moduli space $\A_{11}^{lev}$ is birational to Klein's cubic hypersurface $$\K=V(\sum_{i\in\boldz_5} x_i^2x_{i+1}=0)\subset\Pfour.$$ In particular, $\A_{11}^{lev}$ is unirational but not rational. The cubic hypersurface $\K\subset\P^4$ was first studied by Klein [Kl] (see also [KlF], Band II) in connection with the $z$-embedding of the modular curve $X(11)$ of level 11, which turns out to be defined by the $4\times 4$-minors of the Hessian of the equation of $\K$. In this respect, we note that $\K$ is the unique $\PSL_2(\boldz_{11})$-invariant of degree three in $\Pfour$, and that furthermore $\PSL_2(\boldz_{11})$ is its full automorphism group [Ad1]. The Klein cubic being smooth is unirational but not rational, cf. [CG], [Mur], [Bea]. Our result should be regarded in light of the following facts: ${\A}_t$ is not unirational (and in fact $p_g(\widetilde{\A}_t)\ge 1$) if $t\ge 13$ and $t\ne 14,\; 15,\; 16,\; 18,\; 20,\; 24,\; 30,$ $36$ (Gritsenko [Gri1], [Gri2]), while $\widetilde\A^{lev}_p$ is a 3-fold of general type for all primes numbers $p\ge 37$ (Hulek-Sankaran [HS1], Gritsenko-Hulek, appendix to [Gri1]), where $\widetilde\A^{lev}_p$ is a smooth projective model of a compactification of $\A^{lev}_p$. See also [GH], and the survey paper [HS2] for related results, and [Bo] for a finiteness result in the same spirit. On the other hand, $\overline{\A^{lev}_5}\cong\P(H^0(F_{HM}(3)))$, where bar stands for the Igusa (=Voronoi) toroidal compactification and $F_{HM}$ is the Horrocks-Mumford bundle on $\P^4$ ([HM], [HKW]), while $\A^{lev}_7$ is rational having as birational model a smooth $V_{22}$, a prime Fano 3-fold of index 1 and genus 12 which is rational (see [MS], [Schr] and [GP2] for details). It would be interesting to know how the Klein cubic ``compares'' with the toroidal compactification of $\A^{lev}_{11}$. In a series of forthcoming papers [GP2], [GP3], we will give details as to the structure of $\A_d^{lev}$, $6\le d\le 12$, (excluding $d=9$ and $11$, which are covered here) and $\A_d$, $d=14, 16, 18$ and $20$. In particular, we will prove their rationality or unirationality. Finally, the methods used in this paper also provide an easy proof of the rationality (over $\QQ(\xi)$ for $\xi$ a primitive $9^{\rm th}$ root of unity) of $\A^{lev}_9$. The unirationality of this space also follows implicitly from O'Grady's work [O'G]. He identifies $\A^{lev}_{p^2}$, for $p$ prime, with the moduli space $\A_1(p)$ of pairs of principally polarized abelian surfaces and rank two subspaces of the $p$-torsion points, non-isotropic for the Weil pairing. O'Grady studies the extension to (natural) toroidal compactifications of the finite natural forgetful map $\pi$ from $\A_1(p)$ to the moduli space $\A_1$ of principally polarized abelian surfaces. Not all singularities of the toroidal compactification of $\A_1(p)$ are canonical, so O'Grady needs to describe carefully a partial desingularization all of whose singularities are canonical, before being able to apply Hurwitz's formula for $\pi$ to get an expression for the canonical class. For $p=3$, our method in addition to being simpler also has the advantage of providing an explicit rational parametrization (over $\QQ(\xi)$). {\it Acknowledgments:} We thank Igor Dolgachev, David Eisenbud, Klaus Hulek, and Kristian Ranestad for many useful discussions, and Allan Adler and Gregory Sankaran for a careful reading of a preliminary version of this paper. We are also grateful to Dave Bayer, Dan Grayson and Mike Stillman for {\sl Macaulay} [BS], and {\sl Macaulay2} [GS] which helped us tremendously to understand the shape of the equations described in this paper. The second author also thanks the Mathematical Sciences Research Institute, Berkeley for its hospitality while part of this paper was being written. \section {representations} {Preliminaries} We review basic properties of polarized abelian surfaces, specializing to the case of polarizations of type $(1,11)$. For a more detailed review of this material as used in this paper, see [GP1], \S 1. In general, we use the notation and definitions of [LB] and [Mum]. Let $(A,\L)$ be a general abelian surface with a polarization of type $(1,11)$. Then $|\L|$ induces an embedding of $A\subset\P^{10}=\P(\dual{H^0(\L)})$ as a projectively normal surface (cf. [Laz]) of degree 22 and sectional genus 12. (The projective normality of the general such abelian surface, follows also from the proof of [GP1], Theorem 6.5.) Riemann-Roch tells us that $A$ is contained in $22$ quadrics, which generate the homogeneous ideal of $A$ ([GP1], Theorem 6.5). The line bundle $\L$ induces a natural map from $A$ to its dual, $\phi_{\L}:A\rightarrow \hat A$, given by $x\mapsto t_x^*\L\otimes\L^{-1}$, where $t_x:A\rightarrow A$ is the morphism given by translation by $x\in A$. Its kernel $K(\L)$ is isomorphic to $\boldz_{11}\times\boldz_{11}$, and is dependent only on the polarization. For every $x\in K(\L)$ there is an isomorphism $t_x^*\L \cong \L$. This induces a projective representation $K(\L)\rightarrow {\rm PGL}(H^0(\L))$, which lifts uniquely to a linear representation of $K(\L)$ after taking a central extension of $K(\L)$ $$1 \rTo\, {{\bf C}^*} \rTo\, {\G(\L)} \rTo\, {K(\L)} \rTo\, 0,$$ whose Schur commutator map is the Weil pairing. $\G(\L)$ is the {\it theta group} of $\L$ and is isomorphic to the abstract Heisenberg group $\H(11)$, while the above representation is isomorphic to the Schr\"odinger representation of $\H(11)$ on $V={\bf C}(\boldz_{11})$, the vector space of complex-valued functions on $\boldz_{11}$. An isomorphism between $\G(\L)$ and $\H(11)$, which restricts to the identity on centers induces a symplectic isomorphism between $K(\L)$ and $\boldz_{11}\times\boldz_{11}$. Such an isomorphism is called a {\it level structure of canonical type} on $(A,c_1(\L))$. (See [LB], Chapter 8, \S 3 or [GP1], \S 1.) A decomposition $K(\L)=K_1(\L)\oplus K_2(\L)$, with $K_1(\L)\cong K_2(\L) \cong \boldz_{11}$ subgroups isotropic with respect to the Weil pairing, and a choice of a characteristic $c$ ([LB], Chapter 3, \S 1) for $\L$, define a unique basis $\{\vt^c_x \mid x\in K_1(\L)\}$ of {\it canonical theta functions} for the space $H^0(\L)$ defined in [LB], Chapter 3, \S 2. This basis allows an identification of $H^0(\L)$ with $V$ via $\vt_{\gamma}^c\mapsto x_{\gamma}$, where $x_{\gamma}$ is the function on $\boldz_{11}$ defined by $x_{\gamma}(\delta)=\cases{1&$\gamma=\delta$\cr 0&$\gamma\not=\delta$\cr}$ for $\gamma,\delta\in\boldz_{11}$. The $x_0,\ldots,x_{10}$ can also be identified with coordinates on $\P(\dual{H^0(\L)})$. Under this identification, the representation $\G(\L)\rightarrow {\rm GL}(H^0(\L))$ coincides with the Schr\"odinger representation $\H(11)\rightarrow GL(V)$. We will only consider the action of $\HHH_{11}$, the finite subgroup of $\H(11)\rightarrow GL(V)$ generated in the Schr\"odinger representation by $\sigma$ and $\tau$, where $$ \sigma(x_i)=x_{i-1}, \qquad \tau(x_i)={\xi}^{-i} x_i,$$ for all $i\in{\boldz}_{11}$ and $\xi=e^{2{\pi}i\over 11}$ is a primitive root of unity of order 11. Notice that $\lbrack\sigma,\tau\rbrack=\xi$, so ${\HHH}_{11}$ is a central extension $$1 \rTo\; {\bf\mu_{11}} \rTo\; {\HHH_{11}} \rTo\; {\boldz_{11}}\times{\boldz_{11}} \rTo\; 0.$$ Thus the choice of a canonical level structure means that if $A$ is embedded in $\P(\dual{H^0(\L)})$ using as coordinates $x_{\gamma}=\vt_{\gamma}^c$, $\gamma\in \boldz_{11}$, then the image of $A$ will be invariant under the action of the Heisenberg group $\HHH_{11}$ via the Schr\"odinger representation. (See [LB], Chapter 6, \S 7). If moreover the line bundle $\L$ is chosen to be symmetric (and there are always finitely many choices of such an $\L$ for a given polarization type), then the embedding via $|\L|$ is also invariant under the involution $\iota$, where $$ \iota(x_i)=x_{-i}, \qquad i\in{\boldz}_{11},$$ which restricts to $A$ as the involution $x\mapsto -x$. Let $N(\HHH_{11})$ be the normalizer of $\HHH_{11}$ inside $\SL(V)$, where the inclusion $\HHH_{11}\subset \SL(V)$ is the Schr\"odinger representation. An element $\alpha\in N(\HHH_{11})$ induces an outer automorphism of $\HHH_{11}$, and hence an automorphism of $\boldz_{11}\times\boldz_{11}$ preserving the Weil pairing $e^D$, for $D=(1,11)$. The group of such automorphisms is $\SL_2(\boldz_{11})$, and thus we get a map $\psi : N(\HHH_{11})\rightarrow \SL_2(\boldz_{11})$. As in [HM] \S 1, one sees that the kernel of this map is $\HHH_{11}$ and that $\psi$ is surjective. This leads to extensions \newarrow{Equals}===== $$\diagram[midshaft,small] 1&\rTo &\HHH_{11}&\rTo &N(\HHH_{11})&\rTo^\psi& \SL_2(\boldz_{11})&\rTo& 1\\ & &\dTo & &\dTo & & \dEquals \\ 0&\rTo &\boldz_{11}\times\boldz_{11} &\rTo &N(\HHH_{11})/{\bf{\mu_{11}}} &\rTo& \SL_2(\boldz_{11})&\rTo& 1\\ \enddiagram$$ where in the bottom row $N(\HHH_{11})/{\bf{\mu_{11}}}$ is a semidirect product by the above (symplectic) action of $\SL_2(\boldz_{11})$. Since $H^2(\SL_2(\boldz_{11}),{\bf C}^*)=0$ it follows that $N(\HHH_{11})$ is in fact the semi-direct product $\HHH_{11}\rtimes\SL_2(\boldz_{11})$ (see [HM], \S 1 for details in the identical case of $\HHH_5$). Therefore the Schr\"odinger representation of $\HHH_{11}$ induces an 11 dimensional representation $$\rho_{11}:\SL_2(\boldz_{11})\rTo \SL(V).$$ In terms of generators and relations, cf. [BM], one has: $${\rm PSL}_2(\boldz_{11})=\langle S,T\mid S^{11}={(S^2TS^6T)}^3=1,\quad {(ST)}^3=T^2=1\rangle,$$ where $$S=\pmatrix{1&1\cr 0&1\cr}\qquad {\rm and}\qquad T=\pmatrix{0&\hfill -1\cr 1&\hfill 0\cr},$$ while the representation $\rho_{11}$ is given projectively by $$\rho_{11}(S)=(\xi^{ij/2}\delta_{ij})_{0\le i,j \le 10}\qquad {\rm and}\qquad \rho_{11}(T)={1\over\sqrt{11}}(\xi^{-ij})_{0\le i,j \le 10},$$ where $\xi$ is the above fixed $11^{\rm th}$ root of unity. (See [Tan] and [Si] for details.) The center of $\SL_2(\boldz_{11})$ is generated by $T^2$, and $\rho_{11}(T^2)=-\iota$. Thus the representation $\rho_{11}$ is reducible. In fact, if $V_+$ and $V_-$ are the positive and negative eigenspaces, respectively, of the involution $\iota$ acting on $V$, then $V_+$ and $V_-$ are easily seen to be invariant under $\rho_{11}$, and moreover $\rho_{11}$ splits as $\rho_+\oplus \rho_-$, where $\rho_{\pm}$ is the representation of $\SL_2(\boldz_{11})$ acting on $V_{\pm}$. Note that $\rho_-$ is trivial on the center of $\SL_2(\boldz_{11})$, so it descends to give an irreducible representation $$\rho_-:\PSL_2(\boldz_{11})\rTo \GL(V_-).$$ For the reader's convenience, we reproduce from the Atlas of finite groups [CNPW] the character table for $\PSL_2(\boldz_{11})$: $$ \vbox{\tabskip=0pt \offinterlineskip \halign to 400pt {\hfil#$\;$\vrule&\strut\hfil$\quad\quad$#\hfil&\hfil$\quad\quad$#\hfil&\hfil$\quad\quad$# \hfil&\hfil$\quad\quad$#\hfil&\hfil$\quad\quad$#\hfil& \hfil$\quad\quad$#\hfil&\hfil$\quad\quad$#\hfil&\hfil$\quad\quad$#\hfil\cr Size of conjugacy class&1&55&110&132&132&110&60&60\cr Conjugacy class&$I$& $\gamma_1$&$\gamma_2$&$\gamma_3$&$\gamma_4$& $\gamma_5$&$\gamma_6$&$\gamma_7$\cr Character&&&&&&&&\cr \noalign{\hrule} $\chi_1$&1&1&1&1&1&1&1&1\cr $\chi_2$&5&1&-1&0&0&1&$\beta$&$\bar\beta$\cr $\chi_3$&5&1&-1&0&0&1&$\bar\beta$&$\beta$\cr $\chi_4$&10&-2&1&0&0&1&-1&-1\cr $\chi_5$&10&2&1&0&0&-1&-1&-1\cr $\chi_6$&11&-1&-1&1&1&-1&0&0\cr $\chi_7$&12&0&0&$\alpha$&$\alpha'$&0&1&1\cr $\chi_8$&12&0&0&$\alpha'$&$\alpha$&0&1&1\cr} } $$ where $\alpha={1\over 2}(-1+\sqrt{5}), \alpha'={1\over 2}(-1-\sqrt{5}),$ and $\beta={1\over2}(-1+\sqrt{-11})$. The conjugacy classes are represented by $$ \vbox{\tabskip=0pt \offinterlineskip \halign to 400pt {\strut\hfil#\quad \hfil&\hfil#\quad \hfil&\hfil#\quad \hfil&\hfil#\quad \hfil&\hfil#\quad \hfil& \hfil#\quad \hfil&\hfil#\quad \hfil&\hfil#\hfil\cr $I$&$\gamma_1$&$\gamma_2$&$\gamma_3$&$\gamma_4$& $\gamma_5$&$\gamma_6$&$\gamma_7$\cr $\pmatrix{1&0\cr 0&1}$&$\pmatrix{0&-1\cr 1& 0\cr}$&$\pmatrix{ 1& -1\cr 1& 0\cr}$&$\pmatrix{3&0\cr 0&4\cr}$ &$\pmatrix{5&0\cr 0&9\cr}$&$\pmatrix{3&2\cr 4&3\cr}$&$\pmatrix{1&1\cr 0&1\cr}$&$\pmatrix{1&2\cr 0&1\cr}$\cr}}$$ Similarly, the other direct summand $$\rho_+:\SL_2(\boldz_{11})\rTo \GL(V_+)$$ is one of two mutually dual $6$-dimensional irreducible representations of $\SL_2(\boldz_{11})$. We refer the reader to [Dor] for the character table of $\SL_2(\boldz_{11})$. \section {moduli11} {Moduli of $(1,11)$-polarized abelian surfaces} From now on, let $(A,\L)$ be a general abelian surface with a polarization of type $(1,11)$ and with canonical level structure. As seen in the previous section, $|\L|$ embeds $A\subset\P^{10}=\P(\dual{H^0(\L)})=\P(\dual{V})$ as a projectively normal surface of degree 22 and sectional genus 12, which is invariant under the action of the Heisenberg group $\HHH_{11}$ via the Schr\"odinger representation. In particular, $H^0(\I_A(n))$ is also a representation of weight $n$ of the Heisenberg group (i.e., a central element $z\in{\bf C}^{\ast}$ acts by multiplication with $z^n$), and hence all its irreducible components will have dimension $11/\gcd(11,n)$. (See [LB], pg. 179, for this last fact.) We will first determine equations for the locus of odd two-torsion points of $(1,11)$ polarized abelian surfaces. This is a set in $\P^-=\P(\dual{V_-})$. To analyze the equations which arise, we will need to make use of the $\SL_2(\boldz_{11})$ symmetry present. Notice that $S^2(V)=H^0(\O_{\P(\dual{H^0(\L)})}(2))$ is $66$-dimensional, and as a representation of weight $2$, splits into six isomorphic 11 dimensional representations of $\HHH_{11}$, each isomorphic to a twist $V'$ of the Schr\"odinger representation. On the other hand, by (2.2) of [HM], $\hom_{\HHH_{11}}(V',V\otimes V)$ is an $N(\HHH_{11})/\HHH_{11}\cong \SL_2(\boldz_{11})$-module and coincides with the representation $\rho_{11}$ described in \ref{representations} Decomposing into the positive and negative eigenspaces of the involution $\iota$ we deduce that $$V_+=\hom_{\HHH_{11}}(V',S^2(V))\quad {\rm and}\quad V_-=\hom_{\HHH_{11}}(V',\wedge^2(V)),$$ as $\SL_2(\boldz_{11})$-modules. By (2.2) of [HM], $\varphi: V'\otimes \hom_{\HHH_{11}}(V', S^2(V))\rightarrow S^2(V)$ is an isomorphism of $N(\HHH_{11})$-modules, and since $S^2(V)$ is an irreducible $N(\HHH_{11})$-module it follows that $V_+$ is a six dimensional irreducible representation of $\SL_2(\boldz_{11})$. The $\SL_2(\boldz_{11})$-isomorphism $\varphi: V_+\otimes V'\rightarrow S^2(V)$ can be represented as follows. We use the usual basis $x_0,\ldots,x_{10}$ for $V$ (identified with the basis of canonical theta functions of $H^0(\L)$), and the basis $f_0,\ldots,f_{10}$ of $V'$ such that $\sigma(f_i)=f_{i-1}$ and $\tau(f_i)=\xi^{-2i}f_i$, $i\in \boldz_{11}$. Then there is a basis $e_0,\ldots,e_5$ of $V_+$ (in fact the projection of $x_0,\ldots,x_5$ onto $V_+$ coming from the decomposition $V=V_+\oplus V_-$) such that the map $\varphi$ takes $e_i\otimes f_j$ to the $(i,j)^{\rm th}$ entry of the $6\times 11$ matrix $R_5$, whose entries are $${(R_5)}_{ij}=x_{j+i}x_{j-i},\quad 0\le i \le 5, 0\le j \le 10,$$ where the indices of the variables are $\mod 11$. Thus the span of the entries in any row of $R_5$ are a $\HHH_{11}-$subrepresentation of $S^2(V)$. Also, any $\HHH_{11}-$subrepresentation of $S^2(V)$ can be obtained by taking a linear combination of the rows, and taking the span in $S^2(V)$ of the resulting 11 quadratic polynomials. In addition, if $P\in\P^{10}$ and $v\in V_+$, then $v\cdot R_5(P)=0$ if and only if $P$ is contained in the scheme cut out by the $\HHH_{11}$-subrepresentation of quadrics determined by $v$. Abusing notation, we will also denote by $V_\pm$ the eigenspaces of the involution $\iota$ acting on $V'$ (by $\iota(f_i)=f_{-i}$). Then the restriction of $\varphi$ to $V_+\otimes V_+$ induces a $\SL_2(\boldz_{11})$-isomorphism $$\Phi : \wedge^2(V_+)\rTo S^2(V_-),$$ usually called the {\it intertwining operator} (see [We], [AR] pp. 62--63, 74, or [Ad5] for details). We may regard the intertwining isomorphism $\Phi$ as being induced by a skew-symmetric matrix with entries quadratic polynomials in the coordinates of $V_-$, namely: $\Phi$ takes the element $e_i\wedge e_j$ of $\wedge^2(V_+)$ to the $(i,j)^{\rm th}$ entry of the matrix $$S= \pmatrix{0&{{{x}}_{1}}^{{2}}&{{{x}}_{{2}}}^{{2}}&{{{x}}_{{3}}}^{{2}}& {{{x}}_{{4}}}^{{2}}&{{{x}}_{{5}}}^{{2}}\cr {-{{{x}}_{1}}^{{2}}}&0&{{x}}_{1}{{x}}_{{3}}&{{x}}_{{2}}{{x}}_{{4}}& {{x}}_{{3}}{{x}}_{{5}}&{-{{x}}_{{4}}{{x}}_{{5}}}\cr {-{{{x}}_{{2}}}^{{2}}}&{-{{x}}_{1}{{x}}_{{3}}}&0&{{x}}_{1}{{x}}_{{5}} &{-{{x}}_{{2}}{{x}}_{{5}}}&{-{{x}}_{{3}}{{x}}_{{4}}}\cr {-{{{x}}_{{3}}}^{{2}}}&{-{{x}}_{{2}}{{x}}_{{4}}}&{-{{x}}_{1}{{x}}_{{5}}} &0&{-{{x}}_{1}{{x}}_{{4}}}&{-{{x}}_{{2}}{{x}}_{{3}}}\cr {-{{{x}}_{{4}}}^{{2}}}&{-{{x}}_{{3}}{{x}}_{{5}}}&{{x}}_{{2}}{{x}}_{{5}} &{{x}}_{1}{{x}}_{{4}}&0&{-{{x}}_{1}{{x}}_{{2}}}\cr {-{{{x}}_{{5}}}^{{2}}}&{{x}}_{{4}}{{x}}_{{5}}&{{x}}_{{3}}{{x}}_{{4}} &{{x}}_{{2}}{{x}}_{{3}}&{{x}}_{1}{{x}}_{{2}}&0\cr }. $$ Here we are abusing notation by identifying $x_i\in V$ with the projection of $x_i$ onto $V_-$ under the decomposition $V=V_+\oplus V_-$. Note that $S$ can also be viewed as the restriction of the first $6\times 6$-block of $R_5$ to $\P^-$, where we use $x_1,\ldots,x_5$ as coordinates on $\P^-$ (where $x_{i}=x_{-i}$). We will only need the fact that $S$ arises from the intertwining operator in Lemma 2.1 1) below. The key fact we will need later about $S$ is the simple observation that if $P\in\P^-$, then $v\cdot R_5(P)=0$ if and only if $v\cdot S(P)=0$. Following [GP1], \S 6, we define $D_i\subseteq\P^-$ to be the locus where the matrix $S$ has rank $\le 2i$, for $i=1,2$. By the previous observation, we may interpret $D_i$ as the locus of points in $\P^-\subset\P^{10}=\P(\dual{V})$ which are contained in at least a $(6-2i)$-dimensional family of $\HHH_{11}$-representations of quadrics. Remark also that all the loci $D_i$ are invariant under the action of $\PSL_2(\boldz_{11})$ via the representation $\rho_{-}$ defined in \ref{representations} \lemma{modular} \item{1)} $D_1\subset\P^-$ is a smooth curve of degree 20 and genus 26 isomorphic to the modular curve $X(11)$. (This is Klein's $z$-model of the modular curve $X(11)$. It is the ``trace'' of the origins in the Shioda compactification of elliptic normal curves with level structure in $\P^{10}$, and its embedding is induced by $\lambda^4$, where $\lambda$, a $10^{\rm th}$ root of the canonical bundle on $X(11)$, is the generator of the group of $\PSL_2(\boldz_{11})$-invariant line bundles on $X(11)$. See [Kl], pp. 153-156, [AR] and [Dol] for details). \item{2)} $D_2\subset\P^-$ is an irreducible sextic hypersurface, defined by the $6\times 6$ Pfaffian of $S$. It contains as an open subset the locus in $\P^{10}$ of odd $2$-torsion points of $(1,11)$-polarized abelian surfaces with canonical level structure. \proof: $1)$ Consider the composition $$\Psi:\P^-=\P(\dual{V_-})\rTo \P(S^2(\dual{V_-})) \rTo^{\cong} \P(\wedge^2(\dual{V_+})),$$ where the first map is the Veronese embedding and the second isomorphism is induced by the intertwining operator. Then for $P\in\P^-$, one can write $\Psi(P)=S(P)$, the latter being a skew-symmetric matrix which should be interpreted as an element of $\wedge^2(\dual{V_+})$. It is then clear that $D_1$ is the pull-back under $\Psi$ of the locus in $\P(\wedge^2(\dual{V_+}))$ of rank 2 skew-symmetric matrices. This latter locus can be identified with $\Gr(2,\dual{V_+})$. Thus $D_2$ is isomorphic to the pull-back of $\Gr(2,\dual{V_+})$ under $\Psi$. Adler-Ramanan [AR], Theorem 19.17, study this pull-back. In particular, they show that this pull-back gives the same variety in $\P^-$ as that defined by Klein's equations studied by V\'elu in [Ve]. It is proven in [Ve], (summarized in Th\'eor\`eme 10.6), that Klein's equations give the so-called the $z$-model of $X(11)$, and this model is always nonsingular. (See [Dol], Theorem 5.1 for the explicit statement.) Thus $D_1$ is isomorphic to $X(11)$. The genus of $X(11)$ is well-known: see for example [Hu], pg. 59, for the genus of the modular curve of level $n$. The degree of the $z$-model is calculated in [Ve]. (See also [AR], Corollary 23.28.) $2)$ For a general point $P\in\P^-=\P(V_-)$, $S$ has rank 6, the intertwining operator being an isomorphism, thus $\P^-=D_3\not=D_2$. In fact $D_2\subset\P^-$ is the sextic hypersurface given by the Pfaffian of $S$. For the record, the equation $f_6$ of $D_2$ is $$\eqalign{f_6=-&x_1^2x_2x_3^3+x_1^3x_3x_4^2-x_2^3x_3^2x_5+x_1x_4^3x_5^2 +x_2^2x_4x_5^3\cr &+x_1x_2^4x_4-x_2x_3x_4^4-x_1^4x_2x_5+x_3^4x_4x_5+x_1x_3x_5^4\cr &+x_1x_2x_3^2x_4^2-x_1^2x_2^2x_3x_5-x_1x_2^2x_4^2x_5 -x_1^2x_3x_4x_5^2+x_2x_3^2x_4x_5^2,\cr}$$ though we will not make use of its explicit form. To prove its irreducibility we will use the fact that it is a $\PSL_2(\boldz_{11})$-invariant in $S^6(V_-)$. Without loss of generality we may assume that as a $\PSL_2(\boldz_{11})$ representation $V_-$ has character $\chi_3$. It is easy to see that $\chi_{S^2(V_-)}=\chi_3+\chi_5,$ so in particular there are no $\PSL_2(\boldz_{11})$ invariants in $S^2(V_-)$. On the other hand, $\chi_{S^3(V_-)}=\chi_1+\chi_5+\chi_6+\chi_7,$ so in $S^3(V_-)$ there is precisely one $\PSL_2(\boldz_{11})$-invariant, which we will denote by $f_3$. Thus the only way that $f_6$ could fail to be irreducible is if $f_6=f_3^2$. But $f_6$ is not a square. To see this, set, say $x_4=x_5=0$ in the matrix $S$ and take its Pfaffian, or just set $x_4=x_5=0$ in the above equation for $f_6$. We get only one term, $-x_1^2x_2x_3^3$, which is not a square, so $f_6$ itself cannot be a square, and thus $f_6$ is irreducible. (Alternatively, the irreducibility of $D_2$ follows from [Ad3].) For any point $P\in D_2\setminus D_1$, $S(P)$ is of rank 4, thus $P$ is contained in a pencil of $\HHH_{11}$-subrepresentations of quadrics in $\P^{10}$. The last claim follows now from the fact that $\HHH_{11}\rtimes\langle\iota\rangle$-invariant abelian surfaces in $\P^{10}$ lie on a pencil of $\HHH_{11}$-representations of quadrics, and that odd 2-torsion points of a general abelian surface get mapped to $D_2\setminus D_1\subset\P^-$, and uniquely determine the surface (cf. [GP1], Lemma 6.3 and Lemma 6.4).\Box By \ref{modular}, we may define now the morphism $$\Theta : D_2\setminus D_1\rTo \Gr(2, V_+)=\Gr(2,6)$$ $$D_2\setminus D_1\ni P \rMapsto \ker(S(P))=\{v\in V_+ \mid v\cdot S(P)=0\},$$ which sends a point $P$ to the pencil of $\HHH_{11}$-subrepresentations of quadrics containing it. The general $(1,11)$-polarized abelian surface embedded with level structure via a symmetric line bundle meets $\P^-$ in the (images of the) 6 odd 2-torsion points. By [GP1], Lemma 6.4 they are mapped via $\Theta$ to a single point in $\Gr(2,6)$. Thus $\Theta$ factorizes as a rational map $$\Theta_{11}:\A_{11}^{lev}\rDashto \Gr(2,6)=\Gr(2, V_+),$$ which essentially takes an abelian surface $A$ to the point in $\Gr(2,6)$ corresponding to the $\HHH_{11}$-subrepresentation $H^0(\I_A(2))\subset H^0(\O_{\P(V)}(2))$. \theorem{rat.map11} The map $\Theta_{11}:\A_{11}^{lev}\rDashto \Gr(2,6)$ yields a birational map between $\A_{11}^{lev}$ and $\im(\Theta)$. \proof: This is an immediate consequence of the degeneration arguments in [GP1], Theorem 6.5, and the above results. \Box In order to prove \ref{main} we will need to determine the precise structure of $\im(\Theta)$. To do so, we will use the following representation of the Pl\"ucker embedding of the Grassmannian $\Gr(2,2m)=\Gr(2,W)$, where $W\cong {\bf C}^{2m}$, $m\ge 2$. $\Gr(2,2m)$ is embedded in $\P^{{2m \choose 2}-1}=\P(\wedge^2(W))$ via the Pl\"ucker embedding, as the variety of those $2$-vectors which are totally decomposable. Thus hyperplanes in the Pl\"ucker embedding can be identified with (projectivized) skew symmetric forms $H\in\P(\wedge^2(V)^*)$, and thus with $2m\times 2m$ skew-symmetric matrices. In this setting the Grassmannian $\Gr(2,2m)$ can be also identified (as an embedded variety) with the subvariety $R_1$ of $2m\times 2m$ skew-symmetric matrices of rank two. The hyperplane sections corresponding to points of $R_1$ are the Schubert cycles $\sigma_1$, the sets of lines intersecting a given subspace of codimension 2 in $\P(W)$. More precisely, if $P\in\Gr(2,2m)$ corresponds to a subspace of $W\cong {\bf C}^{2m}$ spanned by the rows of a $2\times 2m$ matrix $$L=\pmatrix{a_1&\cdots&a_{2m}\cr b_1&\cdots&b_{2m}\cr},$$ then the corresponding $2m\times 2m$ skew-symmetric matrix $H_P=(p_{ij})$ has as entries the Pl\"ucker coordinates of $L$ $$p_{ij}=a_ib_j-a_jb_i.$$ This matrix is rank 2, all the rows being linear combinations of the rows of $L$. Conversely, given any $2m\times 2m$ skew-symmetric matrix of rank 2, the span of the rows yields a two-dimensional subspace of ${\bf C}^{2m}=W$, and hence a point in $\Gr(2,2m)$. Furthermore, with obvious abuse of notation, the following correspondence holds: \lemma{chords} For $H\in\P(\wedge^2(W)^*)$ and $k\in\{1,\ldots,m\}$ the following are equivalent \item{1)} $H\in R_k:=\{H\in \P(\wedge^2(W)^*)\mid \rank(H)\le 2k\}$ \item{2)} $H$ lies in the $k$-chordal locus of $R_1\cong\Gr(2,2m)$ (i.e., lies in a linear subspace $\P^{k-1}\subset\P(\wedge^2(W)^*)$ which is $k$-secant to $R_1$). \proof: All these facts are classical, and easy to prove. See for instance [SR].\Box An easy computation shows that the codimension of $R_{m-k}$ in $\P(\wedge^2(W)^*)$ is ${2k\choose 2}$. In particular, $R_{m-1}$ is a hypersurface in $\P(\wedge^2(W)^*)$ of degree $m$, defined by the Pfaffian of the generic skew-symmetric matrix. \remark{orbits} The Pl\"ucker embedding is compatible with the natural action of $\PGL(W)$, and the orbits under this action are exactly $R_k\setminus R_{k-1}$, for $k=\overline{1,m}$, where $R_0=\emptyset$. The following lemma describes the map $\Theta$ in the above setting: \lemma{pfaff} Let $M$ be a $2m\times 2m$ skew-symmetric matrix of forms of rank $2m-2$ on a variety $X$. Then the map $\Theta:X\rightarrow\Gr(2,2m)$ induced by $x\in X\mapsto \ker M(x)\subseteq {\bf C}^{2m}$ is given in (dual) Pl\"ucker coordinates by $$x\rMapsto M^*(x)\in R_1=\Gr(2,2m),$$ where $M^*$ is the $2m\times 2m$ skew-symmetric matrix defined by $$M_{ij}^*=\cases{ (-1)^{i+j}\Pf^{ij}(M)& $ij$,\cr}$$ and where $\Pf^{ij}(M)$ is the Pfaffian of the matrix obtained by deleting the $i^{\rm th}$ and $j^{\rm th}$ rows and columns from $M$. \proof: We'll make use of the following standard facts concerning Pfaffian identities (see [BE] and [Re] for more details). Let $F$ be a free module of rank $n=2m$ over the ring $R$, let $F^\ast$ denote the dual module, and let $f:F^\ast\to F$ be a skew-symmetric morphism (that is the matrix of $f$ corresponding to the choice of a basis in $F$ and the dual basis in $F^\ast$ is a skew-symmetric $n\times n$-matrix $M$). Now, $\wedge F^\ast$ and $\wedge F$ are modules over each other and we adopt here [BE]'s notation in writing $a(b)$ for the result of an operation of $a\in\wedge F$ on $b\in\wedge F^\ast$ and vice-versa; thus $a(b)\in\wedge F^\ast$ and $b(a)\in\wedge F$. The skew-symmetric map $f$ corresponds to an element $\varphi\in \wedge^2F$, such that for all $a^\ast\in F^\ast$ we have $f(a^\ast)=-a^\ast(\varphi)$. In terms of a basis $e_1,\ldots,e_{2m}$ of $F$, if $(f_{ij})$ is the matrix of $f$ with $f_{ij}=-f_{ji}$, then $$\varphi=\sum_{ij$. Thus if the matrix $M^\ast=(m^\ast_{ij})$ is the skew-symmetric matrix with entries $$m^\ast_{ij}= \cases{{(-1)}^{i+j}\Pf^{ij}(M)& if\ $i\ne j$,\cr 0& if\ $i=j$,\cr}$$ then $M^\ast$ is the matrix of $g$ associated with the bases $\{e_i\}$ of $F$ and $\{e_i^\ast\}$ of $F^\ast$. The above compositions read as: $M\cdot M^\ast=0$. Thus also $M^*\cdot M=(M\cdot M^*)^t=0$, as required.\Box We show next that $\overline{\im(\Theta)}$ is a smooth Fano 3-fold of genus 8 and index one. As a corollary of \ref{rat.map11} and \ref{lin.eq} below we obtain then \ref{main}. \theorem{lin.eq} The Zariski closure of $\im(\Theta)\subset\Gr(2,6)=\Gr(2,V_+)$ in the Pl\"ucker embedding has equations given in Pl\"ucker coordinates by $$ p_{23}=-p_{15}, \qquad p_{26}=p_{13}, \qquad p_{14}=-p_{35},\qquad p_{16}=p_{45},\qquad p_{46}=-p_{12}.$$ Furthermore, this linear section of $\Gr(2,6)$ is three-dimensional, smooth, and hence a Fano 3-fold of type $V_{14}$. Furthermore it is birational to the Klein cubic hypersurface $$\K=V(\sum_{i\in\boldz_5} x_i^2x_{i+1}=0)\subset\Pfour.$$ \proof: The first thing to check is that $\im(\Theta)\subset\P^9$ satisfies the five linear relations given above. This can easily be checked by hand using Lemma 2.5 simply by computing the corresponding Pfaffians of the matrix $S$ and showing they satisfy the given relations. Observe also that the $\SL_2(\boldz_{11})$ representation on $\wedge^2(V_+)$ induced by $\rho_+$ decomposes as the sum of a 5 and a 10-dimensional irreducible representations. The equations in the statement of \ref{lin.eq} define this 10-dimensional representation as a subspace of $\wedge^2(V_+)$. To conclude that the closure of the image of $\Theta$ is actually given by these equations, we will show that the subscheme $X$ of $\Gr(2,6)$ defined by these equations is three-dimensional and non-singular, and thus a Fano 3-fold $V_{14}$ of genus 8 and index 1. To this end we will use a classical construction due to G. Fano [Fa], and recast in modern language by Iskovskih [Is1], [Is2] (see also [Pu]), which shows that any $V_{14}$ is birationally equivalent to a (smooth) cubic threefold in $\P^4$. Let $H_1,\ldots,H_5$ be five (linearly independent) hyperplanes in $\P^{14}=\P(H^0(\O_{\Gr(2,6)}(1)))$, and let $$X:=\Gr(2,6)\cap H_1\cap\cdots\cap H_5.$$ By \ref{chords} and the discussion preceding it, we may identify $\dual{(\P^{14})}$ with the space of $6\times 6$ skew-symmetric matrices, and so $\dual{\Gr(2,6)}$ can be naturally identified with the locus $R_2$ of skew-symmetric matrices of rank $\le 4$. As seen above $\dual{\Gr(2,6)}$ is then a cubic hypersurface in $\dual{(\P^{14})}$ defined by the $6\times 6$ Pfaffian of the generic skew-symmetric matrix. Its singular locus is the locus $R_1$ of $6\times 6$ skew-symmetric matrices of rank $\le 2$, which is isomorphic (as an embedded variety) to $\Gr(2,6)$. By \ref{chords}, the Pfaffian cubic is also the secant variety to $R_1\cong\Gr(2,6)$. Let $\P^4:=\langle H_1,\ldots,H_5\rangle \subseteq \dual{(\P^{14})}$ denote the span of the above five hyperplanes as points in $\dual{(\P^{14})}$, and let $$B:=\dual{\Gr(2,6)} \cap \langle H_1,\ldots, H_5 \rangle.$$ Define now a (possibly rational) map $$\Psi :B\rTo \Gr(2,6)$$ $$B\ni b\rMapsto \ker(b)\in \Gr(2,6),$$ where we think of each element $b$ of $B$ as a $6\times 6$ skew-symmetric matrix of rank $\le 4$. Therefore $\Psi$ is defined on all of $B$ iff $B$ is disjoint from the singular locus of $\dual{\Gr(2,6)}$. \lemma{smooth} If the cubic hypersurface $B$ is smooth and $\im(\Psi)\cap X=\emptyset$, then $X$ is a non-singular threefold (Fano of genus 8, index 1). \proof: First, let $x\in X$ be a point where the Zariski tangent space of $X$ at $x$, $T_{X,x}$, has $\dim T_{X,x}>3$. If $l_1,\ldots,l_5$ are the Zariski tangent spaces to $H_1,\ldots, H_5$ at $x$, then $$T_{X,x}=T_{\Gr(2,6),x}\cap l_1\cap \ldots\cap l_5\subseteq T_{ \P^{14},x}.$$ Suppose now that $H_1,\ldots, H_5$ are given by the linear equations $h_1=0,\ldots, h_5=0$, respectively. The only way $T_{X,x}$ could fail to be three dimensional is if there exist a hyperplane section $H$ whose equation is $\sum_{i=1}^5 a_i h_i=0$ for some $a_i$, with $T_{H,x}\supseteq T_{\Gr(2,6),x}$. Thus $H$ must be tangent to $\Gr(2,6)$, and so $H\in B$. If the cubic $B$ is smooth, then $B$ is disjoint from the singular locus of $\dual{\Gr(2,6)}$, and the map $\Psi$ above is defined everywhere on $B$. Now $\Psi(H)$ is the point of the Grassmannian $\Gr(2,6)$ which $H$ is tangent to; thus, in particular, if $H$ is tangent to $\Gr(2,6)$ at a point of $X$, we must have $\Psi(H)\in X$ and $\Psi(B)\cap X\not=\emptyset$. In conclusion, if $\Psi(B)\cap X=\emptyset$, we must have $\dim T_{X,x}=3$ and so $X$ is a non-singular Fano threefold. \Box {\it Proof of \ref{lin.eq} continued:} We use \ref{smooth} to check that the subscheme $X$ of $\Gr(2,6)$ defined by the equations in the statement of \ref{lin.eq} is three-dimensional and non-singular. We can now compute $B$ directly in our case: Let $x_{ij}$, $1\le i