\contentsline {section}{\numberline {1}Introduction}{1}
\contentsline {subsection}{\numberline {1.1}Moduli of stable varieties: the case of surfaces}{1}
\contentsline {subsection}{\numberline {1.2}The problem of families}{2}
\contentsline {subsection}{\numberline {1.3}Canonical models and quotient stacks}{3}
\contentsline {subsection}{\numberline {1.4}Moduli of stacks: the past}{3}
\contentsline {subsection}{\numberline {1.5}Moduli of stacks: this paper}{4}
\contentsline {subsection}{\numberline {1.6}Acknowledgments}{5}
\contentsline {subsection}{\numberline {1.7}Conventions}{5}
\contentsline {subsection}{\numberline {1.8}Notation}{5}
\contentsline {section}{\numberline {2}Cyclotomic stacks and weighted projective stacks}{6}
\contentsline {subsection}{\numberline {2.1}Weighted projective stacks}{6}
\contentsline {subsection}{\numberline {2.2}Closed substacks of weighted projective stacks}{7}
\contentsline {subsection}{\numberline {2.3}Cyclotomic stacks and line bundles}{9}
\contentsline {subsection}{\numberline {2.4}Embedding of a cyclotomic stack in a weighted projective bundle}{13}
\contentsline {section}{\numberline {3}Moduli of stacks with polarizing line bundles}{15}
\contentsline {subsection}{\numberline {3.1}The category ${{\@mathcal S}}ta^{\@mathcal L}$}{15}
\contentsline {subsection}{\numberline {3.2}Normally embeddable stacks}{16}
\contentsline {subsection}{\numberline {3.3}Proof of Theorem 3.1.4\hbox {}}{19}
\contentsline {section}{\numberline {4}Moduli of polarized orbispaces}{20}
\contentsline {subsection}{\numberline {4.1}Orbispaces}{20}
\contentsline {subsection}{\numberline {4.2}Polarizations}{20}
\contentsline {subsection}{\numberline {4.3}Comparison of polarizations}{21}
\contentsline {subsection}{\numberline {4.4}Moduli of canonically polarized orbispaces}{22}
\contentsline {section}{\numberline {5}Koll\'ar families and stacks}{24}
\contentsline {subsection}{\numberline {5.1}Reflexive sheaves, saturation and base change}{24}
\contentsline {subsection}{\numberline {5.2}Koll\'ar families of ${\@mathbb {Q}}$-line bundles}{25}
\contentsline {subsection}{\numberline {5.3}Koll\'ar families and uniformized twisted varieties}{26}
\contentsline {section}{\numberline {6}Semilog canonical singularities and compactifications}{29}
\contentsline {subsection}{\numberline {6.1}Properness results and questions}{30}
\contentsline {section}{\numberline {A}The semilog canonical locus is open}{33}
\contentsline {subsection}{\numberline {A.1}Statement}{33}
\contentsline {subsection}{\numberline {A.2}Constructibility}{34}
\contentsline {subsection}{\numberline {A.3}Proof of Proposition}{34}