\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{section.1}
\contentsline {subsection}{\tocsubsection {}{1.1}{Method of proof}}{5}{subsection.1.1}
\contentsline {subsection}{\tocsubsection {}{1.2}{Related work}}{6}{subsection.1.2}
\contentsline {subsection}{\tocsubsection {}{1.3}{Organisation of the paper}}{6}{subsection.1.3}
\contentsline {subsection}{\tocsubsection {}{1.4}{Acknowledgements}}{6}{subsection.1.4}
\contentsline {section}{\tocsection {}{2}{The non-standard setup}}{7}{section.2}
\contentsline {subsection}{\tocsubsection {}{2.1}{Coarse pseudo-finite dimension}}{7}{subsection.2.1}
\contentsline {subsection}{\tocsubsection {}{2.2}{Incidence bounds and Szemer\'edi-Trotter}}{15}{subsection.2.2}
\contentsline {section}{\tocsection {}{3}{Warm-up: the Elekes-Szab\'o theorem}}{17}{section.3}
\contentsline {section}{\tocsection {}{4}{Necessity of general position}}{26}{section.4}
\contentsline {section}{\tocsection {}{5}{Projective geometries arising from varieties without power-saving}}{27}{section.5}
\contentsline {subsection}{\tocsubsection {}{5.1}{Geometries and modularity}}{27}{subsection.5.1}
\contentsline {subsection}{\tocsubsection {}{5.2}{Coarse general position, coherence and modularity}}{29}{subsection.5.2}
\contentsline {section}{\tocsection {}{6}{Varieties with coherent generics}}{33}{section.6}
\contentsline {section}{\tocsection {}{7}{Asymptotic consequences}}{34}{section.7}
\contentsline {subsection}{\tocsubsection {}{7.1}{Sharpness}}{36}{subsection.7.1}
\contentsline {subsection}{\tocsubsection {}{7.2}{Proofs of the main results}}{39}{subsection.7.2}
\contentsline {section}{\tocsection {}{8}{Coherence in subgroups}}{40}{section.8}
\contentsline {section}{\tocsection {Appendix}{A}{Projective geometries fully embedded in algebraic geometry}}{42}{appendix.A}
\contentsline {section}{\tocsection {Appendix}{}{References}}{47}{section*.3}