\contentsline {section}{\tocsection {}{1}{Introduction and Preliminaries}}{1}
\contentsline {section}{\tocsection {}{2}{Hilbert-Mumford index of Hilbert points}}{3}
\contentsline {subsection}{\tocsubsection {}{2.1}{A Gr\"obner basis algorithm for computing the Hilbert-Mumford index}}{4}
\contentsline {subsection}{\tocsubsection {}{2.2}{Macaulay 2 implementation}}{8}
\contentsline {subsection}{\tocsubsection {}{2.3}{State polytopes}}{9}
\contentsline {section}{\tocsection {}{3}{Applications}}{11}
\contentsline {subsection}{\tocsubsection {}{3.1}{The rational bicuspidal curve of genus two}}{11}
\contentsline {subsection}{\tocsubsection {}{3.2}{Degeneration of cuspidal curves}}{14}
\contentsline {subsubsection}{\tocsubsubsection {}{3.2.1}{Flat limit of $\rho (\alpha ).[C_1]$}}{15}
\contentsline {subsection}{\tocsubsection {}{3.3}{Hilbert unstable curves - Instability of elliptic bridges}}{18}
\contentsline {section}{\tocsection {}{}{References}}{25}