\contentsline {section}{\tocsection {}{1}{Introduction}}{2}
\contentsline {section}{\tocsection {}{2}{Statement of results and strategy of proof}}{5}
\contentsline {subsection}{\tocsubsection {}{2.1}{Stability notions for algebraic curves}}{5}
\contentsline {subsection}{\tocsubsection {}{2.2}{Construction of the small contraction $\Psi $}}{7}
\contentsline {subsection}{\tocsubsection {}{2.3}{Construction of the flip $\Psi ^+$}}{11}
\contentsline {subsection}{\tocsubsection {}{2.4}{Stability results on bicanonical curves}}{12}
\contentsline {subsection}{\tocsubsection {}{2.5}{Detailed roadmap for the GIT analysis}}{13}
\contentsline {section}{\tocsection {}{3}{GIT of Chow varieties and Hilbert schemes}}{14}
\contentsline {subsection}{\tocsubsection {}{3.1}{GIT of Chow points}}{15}
\contentsline {subsection}{\tocsubsection {}{3.2}{GIT of Hilbert points}}{16}
\contentsline {subsection}{\tocsubsection {}{3.3}{Polarizations on Hilbert schemes}}{17}
\contentsline {subsection}{\tocsubsection {}{3.4}{Tautological classes on the Hilbert scheme}}{18}
\contentsline {subsection}{\tocsubsection {}{3.5}{Hilbert points and Hilbert schemes}}{19}
\contentsline {subsection}{\tocsubsection {}{3.6}{Chow stability and Hilbert stability}}{20}
\contentsline {subsection}{\tocsubsection {}{3.7}{Filtered Hilbert polynomials}}{21}
\contentsline {subsection}{\tocsubsection {}{3.8}{Hilbert schemes of curves}}{22}
\contentsline {section}{\tocsection {}{4}{Basin of attraction and equivalences}}{23}
\contentsline {section}{\tocsection {}{5}{Computations over the moduli space of stable curves}}{26}
\contentsline {section}{\tocsection {}{6}{Properties of c-semistable and h-semistable curves}}{30}
\contentsline {subsection}{\tocsubsection {}{6.1}{Embedding c-semistable curves}}{30}
\contentsline {subsection}{\tocsubsection {}{6.2}{Basic properties of tacnodal curves}}{34}
\contentsline {section}{\tocsection {}{7}{Unstable bicanonical curves}}{36}
\contentsline {subsection}{\tocsubsection {}{7.1}{Badly singular curves are Chow unstable}}{36}
\contentsline {subsection}{\tocsubsection {}{7.2}{Polarizations on semistable limits of bicanonical curves}}{38}
\contentsline {subsection}{\tocsubsection {}{7.3}{Elliptic subcurves meeting the rest of the curve in one point}}{39}
\contentsline {subsection}{\tocsubsection {}{7.4}{Hilbert unstable curves}}{41}
\contentsline {section}{\tocsection {}{8}{Classification of curves with automorphisms}}{41}
\contentsline {subsection}{\tocsubsection {}{8.1}{Rosaries}}{42}
\contentsline {subsection}{\tocsubsection {}{8.2}{Classification of automorphisms}}{44}
\contentsline {section}{\tocsection {}{9}{Interpreting the flip via GIT}}{46}
\contentsline {section}{\tocsection {}{10}{Stability under one-parameter subgroups}}{49}
\contentsline {subsection}{\tocsubsection {}{10.1}{Stability analysis: Open rosaries}}{49}
\contentsline {subsection}{\tocsubsection {}{10.2}{Basin of attraction: Open rosaries}}{53}
\contentsline {subsection}{\tocsubsection {}{10.3}{Stability analysis: Closed rosaries}}{54}
\contentsline {subsection}{\tocsubsection {}{10.4}{Basin of attraction: Closed rosaries}}{57}
\contentsline {subsection}{\tocsubsection {}{10.5}{Stability analysis: Closed rosaries with a broken bead}}{57}
\contentsline {subsection}{\tocsubsection {}{10.6}{Basin of attraction: Closed rosary with a broken bead}}{60}
\contentsline {section}{\tocsection {}{11}{Proofs of semistability and applications}}{61}
\contentsline {subsection}{\tocsubsection {}{11.1}{Elliptic bridges and their replacements}}{61}
\contentsline {subsection}{\tocsubsection {}{11.2}{Chow semistability of c-semistable curves}}{67}
\contentsline {subsection}{\tocsubsection {}{11.3}{ Hilbert semistability of h-semistable curves}}{68}
\contentsline {section}{\tocsection {}{}{References}}{73}