\relax \citation{AOV} \citation{A-V} \citation{A-V} \citation{logtwisted} \citation{homstack} \citation{bounded} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.1}{Context and statement of main result}}{1}} \citation{A-V} \citation{logtwisted} \citation{homstack} \citation{bounded} \citation{A-V} \citation{ACV} \citation{logtwisted} \citation{homstack} \citation{bounded} \citation{AOV} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.2}{The paper is organized as follows}}{2}} \citation{Raynaud} \citation{raynaudgroup} \citation{Romagny} \citation{Tossici-actions} \citation{raynaudgroup} \citation{Aoki} \citation{Aokierratum} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.3}{The question of the correct generality}}{3}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.7}{Acknowledgements}}{4}} \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Twisted curves}}{4}} \newlabel{section4}{{2}{4}} \newlabel{A1}{{2.1}{4}} \citation{AOV} \citation{D-M} \citation{AOV} \newlabel{Prop:2.3}{{2.3}{5}} \newlabel{localiso}{{2.3.1}{6}} \newlabel{goodaction}{{2.3.2}{6}} \citation{L-MB} \citation{Olsson-sheaves} \citation{Olsson-sheaves} \citation{Laszlo-Olsson} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.4}{The cotangent complex and an alternative proof of (v)}}{7}} \citation{A-V} \newlabel{cohlemb}{{2.6}{9}} \citation{Artin} \citation{Brochard} \citation{AOV} \newlabel{Picmapb}{{2.7.1}{10}} \newlabel{cokernel}{{2.8}{10}} \citation{Keel-Mori} \citation{Conrad} \newlabel{Def:Pic^o}{{2.11}{12}} \newlabel{3.7}{{2.12}{12}} \@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Interlude: Relative moduli spaces}}{12}} \newlabel{interlude}{{3}{12}} \newlabel{Th:rel-cms}{{3.1}{12}} \citation{AOV} \newlabel{Prop:B3}{{3.4}{13}} \newlabel{relative-example}{{3.5}{13}} \citation{AOV} \@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Twisted stable maps}}{15}} \newlabel{Section:3}{{4}{15}} \newlabel{Prop:existence}{{4.1}{16}} \newlabel{Lem:relative-algebraic}{{4.2}{16}} \citation{AGV} \newlabel{Th:3.1}{{4.3}{17}} \newlabel{compositemap}{{4.3.1}{17}} \newlabel{5.2}{{4.4}{17}} \citation{AOV} \citation{bounded} \citation{homstack} \citation{homstack} \newlabel{Prop:valuative}{{4.5}{19}} \newlabel{Lem:purity}{{4.6}{19}} \citation{AOV} \citation{AOV} \citation{AOV} \@writefile{toc}{\contentsline {section}{\tocsection {}{5}{Reduction of Spaces of Galois admissible covers}}{22}} \newlabel{reductionsection}{{5}{22}} \citation{ACV} \citation{ACV} \newlabel{flattheorem}{{5.1}{23}} \newlabel{flatmap}{{5.2.1}{23}} \citation{homstack} \citation{SGA4} \newlabel{flattheorembis}{{5.3}{24}} \newlabel{Sec:G-tame-etale}{{5.4}{24}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.4}{The case when $G$ is a tame \'etale group scheme}}{24}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.5}{The case when $G$ is locally diagonalizable}}{24}} \newlabel{Lem:tors-picx}{{5.7}{25}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.9}{Observations on fixed points}}{25}} \newlabel{Aseq}{{5.9.1}{25}} \newlabel{smoothlem}{{5.10}{26}} \newlabel{1.5.6}{{5.11.1}{27}} \newlabel{5.9}{{5.12}{27}} \newlabel{shortseq}{{5.12.1}{27}} \newlabel{map1}{{5.12.2}{27}} \citation{L-MB} \newlabel{kermap}{{5.13.1}{28}} \newlabel{cohlem}{{5.15}{28}} \newlabel{cohgroups}{{5.15.1}{29}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.16}{General $G$: setup}}{29}} \newlabel{flatmap2}{{5.16.1}{29}} \newlabel{relflat}{{5.16.2}{29}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.17}{Reduction to $\mathcal C'$ connected.}}{30}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.18}{General $G$: strategy}}{31}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.19}{Structure of $\relax $\@@underline {\hbox {\text {\rm Sec}}}\mathsurround \z@ $\relax _T (\mathcal G'/\mathcal C')$ and $\relax $\@@underline {\hbox {\text {\rm Sec}}}\mathsurround \z@ $\relax _{\mathcal {B}H_T} (\mathcal G/\mathcal C)$}}{31}} \newlabel{Lem:sec-BH-gerbe}{{5.20}{31}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.22}{The pushforward of $\text {Pic}_{\mathcal C/\mathcal {B}H_T}[\mathbf {\EuScript {X}}]$ and its torsor ${\text {\rm Sec}}_{\mathcal {B}H_T} (\mathcal G/\mathcal C)$}}{32}} \newlabel{1.5.1}{{5.23}{32}} \newlabel{torsorlem}{{5.24}{33}} \newlabel{Prop:sec-T-gerbe}{{5.27}{33}} \citation{DR} \@writefile{toc}{\contentsline {section}{\tocsection {}{6}{Example: reduction of $\mathbf {\EuScript {X}}(2)$ in characteristic 2}}{34}} \newlabel{examplesection}{{6}{34}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{6.1}{$\mathbf {\EuScript {X}}(2)$ as a distinguished component in $\mathcal {K}_{0,4}(\mathcal {B}{\boldsymbol {\mu }}_2)$}}{34}} \citation{AGV} \citation{AGV} \citation{AOV} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{6.2}{Cyclotomic inertia}}{35}} \newlabel{grouphom}{{6.4}{35}} \citation{AGV} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{6.5}{Rigidified cyclotomic inertia}}{36}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{6.6}{Evaluation maps}}{36}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{6.7}{Back to $\mathcal {K}\subset \mathcal {K}_{0,4}(\mathcal {B}{\boldsymbol {\mu }}_2)$}}{37}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{6.8}{What does $\mathcal {K}_{\mathbb {F}_2}$ parametrize?}}{37}} \citation{KM} \citation{KM} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{6.9}{$\mathbf {\EuScript {X}}(2)$ as a distinguished component in $\mathcal {K}_{1,1}(\mathcal {B}{\boldsymbol {\mu }}_2^2)$.}}{38}} \newlabel{KMinclusion}{{6.9.1}{38}} \newlabel{coprod}{{6.9.2}{39}} \newlabel{fibermap}{{6.9.3}{39}}