\contentsline {section}{\tocsection {}{1}{Introduction}}{1}
\contentsline {section}{\tocsection {}{2}{Guiding questions and examples}}{3}
\contentsline {section}{\tocsection {}{3}{Background results}}{5}
\contentsline {section}{\tocsection {}{4}{Deformation results for stable maps}}{6}
\contentsline {section}{\tocsection {}{5}{K3 surfaces over finite fields}}{8}
\contentsline {section}{\tocsection {}{6}{Curves on K3 surfaces in positive and mixed characteristic}}{10}
\contentsline {section}{\tocsection {}{7}{Proof of the Main Theorem}}{12}
\contentsline {section}{\tocsection {}{}{References}}{14}