\contentsline {section}{\numberline {1}Introduction}{3}
\contentsline {subsection}{\numberline {1.1}Fano 3-folds}{3}
\contentsline {subsection}{\numberline {1.2}Reminder}{3}
\contentsline {paragraph}{Curves:}{3}
\contentsline {paragraph}{Surfaces:}{3}
\contentsline {subsection}{\numberline {1.3}Iskovskikh's table of Fano 3-folds}{4}
\contentsline {subsection}{\numberline {1.4}Boundedness}{4}
\contentsline {subsection}{\numberline {1.5}A preview of the method}{5}
\contentsline {section}{\numberline {2}Basic properties of Gorenstein Fano 3-folds}{6}
\contentsline {subsection}{\numberline {2.1}Free linear system}{6}
\contentsline {subsection}{\numberline {2.2}The projective model of a K3 surface}{7}
\contentsline {subsection}{\numberline {2.3}The anticanonical model of a Fano 3-fold}{9}
\contentsline {subsection}{Appendix A: Varieties of minimal degree}{9}
\contentsline {subsection}{\numberline {2.4}Defining equations of $X_{2g-2}$}{10}
\contentsline {section}{\numberline {3}Varieties with canonical curve sections}{12}
\contentsline {subsection}{\numberline {3.1}Results of the 1982 Warwick calculations}{12}
\contentsline {subsection}{\numberline {3.2}Main aim}{13}
\contentsline {section}{\numberline {4}The $G_2$-manifold $\Sigma _{18}^5\subset \@mathbb P^{13}$}{15}
\contentsline {subsection}{\numberline {4.1}Homogenous space}{15}
\contentsline {subsection}{\numberline {4.2}General theory, the Borel--Weil--Bott theorem}{17}
\contentsline {subsection}{\numberline {4.3}Topology of the $G_2$ manifold}{17}
\contentsline {subsection}{\numberline {4.4}Null octonion Grassmann variety}{19}
\contentsline {subsection}{\numberline {4.5}Trivectors in 7-space and the $G_2$-manifold}{20}
\contentsline {subsection}{\numberline {4.6}The $G_2$-manifold in the 7 dimensional orthogonal Grassmann in $\Grass (2,7)$}{22}
\contentsline {subsection}{\numberline {4.7}Zero locus of a degenerate spinor and degeneration of $\Sigma _{18}^5$}{24}
\contentsline {subsection}{\numberline {4.8}Hyperplane section of Schubert type}{26}
\contentsline {section}{\numberline {5}The linear section theorem (case $g=10$)}{27}
\contentsline {subsection}{\numberline {5.1}Generality conditions on a polarized K3 surface}{28}
\contentsline {subsection}{\numberline {5.2}Generality and gonality of a K3 surface}{28}
\contentsline {subsection}{\numberline {5.3}Case division}{29}
\contentsline {subsection}{\numberline {5.4}Construction of vector bundle in Case\nobreakspace {}D}{29}
\contentsline {subsection}{\numberline {5.5}Proof of linear section theorem in the non D case}{30}
\contentsline {paragraph}{Construction of $\@mathcal E$}{30}
\contentsline {paragraph}{Proof}{32}
\contentsline {paragraph}{Proof}{34}
\contentsline {paragraph}{Proof}{34}
\contentsline {section}{\numberline {6}The Lefschetz principle}{35}
\contentsline {subsection}{\numberline {6.1}Bott vanishing}{35}
\contentsline {subsection}{\numberline {6.2}Twistings of the tangent bundle}{36}
\contentsline {subsection}{\numberline {6.3}The Lefschetz principle}{36}
\contentsline {paragraph}{Notation}{37}
\contentsline {paragraph}{\sc Case: $\alpha $ is surjective}{38}
\contentsline {paragraph}{\sc Case: $\alpha $ is not surjective}{38}
\contentsline {section}{\numberline {7}Indecomposable Fano 3-fold}{38}
\contentsline {subsection}{\numberline {7.1}The generality of a K3 surface $S\in |{-}K_X|$ on an indecomposable $X$}{38}
\contentsline {paragraph}{Proof}{39}
\contentsline {subsection}{\numberline {7.2}The bound on the genus}{39}
\contentsline {paragraph}{Proof}{40}
\contentsline {section}{References}{40}