\contentsline {section}{\numberline {1}Introduction}{2}
\contentsline {section}{\numberline {2}Fano Threefolds}{6}
\contentsline {section}{\numberline {3}A Chow ring calculation}{9}
\contentsline {section}{\numberline {4}The argument via monodromy}{14}
\contentsline {subsection}{\numberline {4.1}Generalities about quartic K3's}{15}
\contentsline {subsection}{\numberline {4.2}Analysis of the points of $C_H \cap L$}{17}
\contentsline {section}{\numberline {5}Rational points on quartic surfaces}{20}
\contentsline {section}{\numberline {6}An example: the Fermat quartic}{24}
\contentsline {section}{\numberline {7}Quartic threefolds}{26}
\contentsline {section}{\numberline {8}Other elliptic surfaces}{29}