\relax \citation{I1} \citation{I2} \citation{I2} \@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{3}} \newlabel{sec!intro}{{1}{3}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Fano 3-folds}{3}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Reminder}{3}} \@writefile{toc}{\contentsline {paragraph}{Curves:}{3}} \@writefile{toc}{\contentsline {paragraph}{Surfaces:}{3}} \citation{IS} \@writefile{toc}{\contentsline {subsection}{\numberline {1.3}Iskovskikh's table of Fano 3-folds}{4}} \@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Cases 6--10 and 12 are varieties $V^3_{2g-2}\subset \@mathbb P^{g+1}$.}}{4}} \newlabel{tab!IS}{{1}{4}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.4}Boundedness}{4}} \citation{R1} \citation{YPG} \@writefile{toc}{\contentsline {subsection}{\numberline {1.5}A preview of the method}{5}} \citation{SD} \citation{M} \@writefile{toc}{\contentsline {section}{\numberline {2}Basic properties of Gorenstein Fano 3-folds}{6}} \newlabel{sec!basic}{{2}{6}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Free linear system}{6}} \newlabel{eq!monog}{{2.1}{6}} \newlabel{df!indec}{{2.5}{7}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.2}The projective model of a K3 surface}{7}} \newlabel{ssec!K3emb}{{2.2}{7}} \citation{SD} \@writefile{toc}{\contentsline {subsection}{\numberline {2.3}The anticanonical model of a Fano 3-fold}{9}} \newlabel{ssec!double}{{2.3}{9}} \@writefile{toc}{\contentsline {subsection}{Appendix A: Varieties of minimal degree}{9}} \citation{ACGH} \citation{GL} \@writefile{toc}{\contentsline {subsection}{\numberline {2.4}Defining equations of $X_{2g-2}$}{10}} \citation{Mu1} \citation{Mu2} \citation{Wi} \citation{Me} \@writefile{toc}{\contentsline {section}{\numberline {3}Varieties with canonical curve sections}{12}} \newlabel{sec!cc}{{3}{12}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Results of the 1982 Warwick calculations}{12}} \citation{MU} \@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces Projective homogenous spaces of coindex $3$}}{13}} \newlabel{tab!4cs}{{2}{13}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The Dynkin diagrams $E_8$, $E_7$, $F_4$ and $G_2$ and their completions}}{13}} \newlabel{fig!4cs}{{1}{13}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Main aim}{13}} \newlabel{exa!sch}{{3.2}{14}} \newlabel{eq!2+4}{{3.1}{14}} \newlabel{eq!sch1}{{3.2}{14}} \citation{Ba} \newlabel{eq!sch2}{{3.3}{15}} \@writefile{toc}{\contentsline {section}{\numberline {4}The $G_2$-manifold $\Sigma _{18}^5\subset \@mathbb P^{13}$}{15}} \newlabel{sec!G2}{{4}{15}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Homogenous space}{15}} \newlabel{ssec!homog}{{4.1}{15}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The root system of type $G_2$}}{15}} \newlabel{fig!G2}{{2}{15}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.2}General theory, the Borel--Weil--Bott theorem}{17}} \newlabel{ssec!BBW}{{4.2}{17}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.3}Topology of the $G_2$ manifold}{17}} \newlabel{result!G2_flag_decomp}{{4.3}{17}} \@writefile{lot}{\contentsline {table}{\numberline {3}{\ignorespaces The 4 projective homogeneous spaces of coindex 3}}{18}} \newlabel{tab!HSym}{{3}{18}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.4}Null octonion Grassmann variety}{19}} \newlabel{ssec!8Gr}{{4.4}{19}} \newlabel{eq!8vs}{{4.1}{19}} \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces $\@mathbb P^2(\@mathbb F_2)$ and multiplication of unit octonions}}{19}} \newlabel{fig!seven}{{3}{19}} \newlabel{eq!alt}{{4.2}{20}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.5}Trivectors in 7-space and the $G_2$-manifold}{20}} \newlabel{ssec!3vect&}{{4.5}{20}} \newlabel{eq!si}{{4.3}{20}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.6}The $G_2$-manifold in the 7 dimensional orthogonal Grassmann in $\Grass (2,7)$}{22}} \newlabel{ssec!G2in7Gr}{{4.6}{22}} \newlabel{lem!ioG}{{4.4}{22}} \newlabel{eq!star}{{4.4}{23}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.7}Zero locus of a degenerate spinor and degeneration of $\Sigma _{18}^5$}{24}} \newlabel{ssec!degen}{{4.7}{24}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.8}Hyperplane section of Schubert type}{26}} \newlabel{ssec!hyp}{{4.8}{26}} \newlabel{prop!isol}{{4.10}{27}} \@writefile{toc}{\contentsline {section}{\numberline {5}The linear section theorem (case $g=10$)}{27}} \newlabel{sec!g10}{{5}{27}} \newlabel{thm!g10}{{5.1}{27}} \citation{M} \citation{Mu2} \@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Generality conditions on a polarized K3 surface}{28}} \newlabel{ssec!genK3}{{5.1}{28}} \newlabel{df!BN}{{5.2}{28}} \newlabel{Note!Moish}{{5.3}{28}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Generality and gonality of a K3 surface}{28}} \newlabel{ssec!gon}{{5.2}{28}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Case division}{29}} \newlabel{ssec!CD}{{5.3}{29}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.4}Construction of vector bundle in Case\nobreakspace {}D}{29}} \newlabel{ssec!Dbun}{{5.4}{29}} \citation{K} \citation{KL1} \citation{KL2} \@writefile{toc}{\contentsline {subsection}{\numberline {5.5}Proof of linear section theorem in the non D case}{30}} \newlabel{ssec!NDpf}{{5.5}{30}} \@writefile{toc}{\contentsline {paragraph}{Construction of $\@mathcal E$}{30}} \newlabel{eq!1}{{5.1}{31}} \newlabel{eq!2}{{5.2}{31}} \@writefile{toc}{\contentsline {paragraph}{Proof}{32}} \newlabel{eq!spl}{{5.3}{32}} \newlabel{rem!CD}{{5.7}{32}} \newlabel{lem!fr}{{5.9}{33}} \citation{Fu} \@writefile{toc}{\contentsline {paragraph}{Proof}{34}} \@writefile{toc}{\contentsline {paragraph}{Proof}{34}} \citation{Bo} \newlabel{rmk!CartD}{{5.16}{35}} \@writefile{toc}{\contentsline {section}{\numberline {6}The Lefschetz principle}{35}} \newlabel{sec!Lefsch}{{6}{35}} \@writefile{toc}{\contentsline {subsection}{\numberline {6.1}Bott vanishing}{35}} \newlabel{ssec!Bott}{{6.1}{35}} \@writefile{toc}{\contentsline {subsection}{\numberline {6.2}Twistings of the tangent bundle}{36}} \newlabel{ssec!twtgt}{{6.2}{36}} \@writefile{toc}{\contentsline {subsection}{\numberline {6.3}The Lefschetz principle}{36}} \newlabel{ssec!Lef}{{6.3}{36}} \@writefile{toc}{\contentsline {paragraph}{Notation}{37}} \@writefile{toc}{\contentsline {paragraph}{\sc Case: $\alpha $ is surjective}{38}} \@writefile{toc}{\contentsline {paragraph}{\sc Case: $\alpha $ is not surjective}{38}} \@writefile{toc}{\contentsline {section}{\numberline {7}Indecomposable Fano 3-fold}{38}} \newlabel{sec!F3f}{{7}{38}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.1}The generality of a K3 surface $S\in |{-}K_X|$ on an indecomposable $X$}{38}} \newlabel{ssec!gen_antic_K3}{{7.1}{38}} \@writefile{toc}{\contentsline {paragraph}{Proof}{39}} \newlabel{eq!rO>a}{{7.1}{39}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.2}The bound on the genus}{39}} \newlabel{ssec!bd_on_g}{{7.2}{39}} \bibcite{ACGH}{ACGH} \bibcite{Ba}{Ba} \@writefile{toc}{\contentsline {paragraph}{Proof}{40}} \@writefile{toc}{\contentsline {section}{References}{40}} \bibcite{B1}{B1} \bibcite{B2}{B2} \bibcite{Bo}{Bo} \bibcite{Bu}{Bu} \bibcite{Fu}{Fu} \bibcite{GL}{GL} \bibcite{I1}{I1} \bibcite{I2}{I2} \bibcite{IS}{IS} \bibcite{K}{K} \bibcite{KL1}{KL1} \bibcite{KL2}{KL2} \bibcite{Me}{Me} \bibcite{M}{M} \bibcite{MM}{MM} \bibcite{Mu1}{Mu1} \bibcite{Mu2}{Mu2} \bibcite{Mu3}{Mu3} \bibcite{Mu4}{Mu4} \bibcite{Mu5}{Mu5} \bibcite{Mu7}{Mu7} \bibcite{Mu8}{Mu8} \bibcite{Mu9}{Mu9} \bibcite{Mu10}{Mu10} \bibcite{Mu11}{Mu11} \bibcite{Mu12}{Mu12} \bibcite{Mu13}{Mu13} \bibcite{Mu14}{Mu14} \bibcite{MU}{MU} \bibcite{R1}{R1} \bibcite{YPG}{YPG} \bibcite{SD}{SD} \bibcite{Wi}{Wi}