\contentsline {section}{\tocsection {}{}{Introduction}}{1}
\contentsline {subsection}{\tocsubsection {}{0.1}{Gromov--Witten invariants in the smooth case}}{1}
\contentsline {subsection}{\tocsubsection {}{0.2}{Invariants of pairs and degenerate varieties}}{2}
\contentsline {subsection}{\tocsubsection {}{0.3}{The twisting method}}{3}
\contentsline {subsection}{\tocsubsection {}{0.4}{The degeneration formula}}{4}
\contentsline {subsubsection}{\tocsubsubsection {}{0.4.2}{User's guide - left hand side}}{4}
\contentsline {paragraph}{\tocparagraph {}{}{1}}{4}
\contentsline {paragraph}{\tocparagraph {}{}{2}}{4}
\contentsline {paragraph}{\tocparagraph {}{}{3}}{4}
\contentsline {paragraph}{\tocparagraph {}{}{4}}{4}
\contentsline {paragraph}{\tocparagraph {}{}{5}}{4}
\contentsline {paragraph}{\tocparagraph {}{}{6}}{5}
\contentsline {paragraph}{\tocparagraph {}{}{7}}{5}
\contentsline {paragraph}{\tocparagraph {}{}{8}}{5}
\contentsline {paragraph}{\tocparagraph {}{}{9}}{5}
\contentsline {subsubsection}{\tocsubsubsection {}{0.4.3}{Right hand side}}{5}
\contentsline {paragraph}{\tocparagraph {}{}{1}}{5}
\contentsline {paragraph}{\tocparagraph {}{}{2}}{5}
\contentsline {paragraph}{\tocparagraph {}{}{3}}{5}
\contentsline {paragraph}{\tocparagraph {}{}{4}}{5}
\contentsline {paragraph}{\tocparagraph {}{}{5}}{5}
\contentsline {paragraph}{\tocparagraph {}{}{6}}{5}
\contentsline {paragraph}{\tocparagraph {}{}{7}}{5}
\contentsline {paragraph}{\tocparagraph {}{}{8}}{5}
\contentsline {paragraph}{\tocparagraph {}{}{9}}{5}
\contentsline {paragraph}{\tocparagraph {}{}{10}}{5}
\contentsline {subsection}{\tocsubsection {}{0.5}{The logarithmic approach}}{6}
\contentsline {subsection}{\tocsubsection {}{0.6}{Further directions}}{6}
\contentsline {subsection}{\tocsubsection {}{0.7}{Outline of the paper}}{7}
\contentsline {subsection}{\tocsubsection {}{0.8}{Conventions}}{8}
\contentsline {subsection}{\tocsubsection {}{0.9}{Notation}}{8}
\contentsline {subsection}{\tocsubsection {}{}{Acknowledgements}}{9}
\contentsline {section}{\tocsection {}{1}{Roots and twists}}{9}
\contentsline {subsection}{\tocsubsection {}{1.1}{Twisted curves}}{9}
\contentsline {subsubsection}{\tocsubsubsection {}{1.1.1}{Automorphism groups of twisted curves}}{9}
\contentsline {subsubsection}{\tocsubsubsection {}{1.1.2}{Deformations of twisted curves}}{10}
\contentsline {subsubsection}{\tocsubsubsection {}{1.1.3}{Comparing deformations and automorphisms of ${\mathcal C}$ and $C$}}{10}
\contentsline {subsubsection}{\tocsubsubsection {}{1.1.4}{Moduli of twisted curves}}{11}
\contentsline {subsection}{\tocsubsection {}{1.2}{Root stacks}}{11}
\contentsline {subsubsection}{\tocsubsubsection {}{1.2.1}{Power morphisms}}{11}
\contentsline {subsubsection}{\tocsubsubsection {}{1.2.2}{Root stack of a line bunle}}{11}
\contentsline {subsubsection}{\tocsubsubsection {}{1.2.3}{Root of a divisor}}{12}
\contentsline {subsubsection}{\tocsubsubsection {}{1.2.4}{Roots with several divisors}}{12}
\contentsline {subsubsection}{\tocsubsubsection {}{1.2.5}{Comparison of roots}}{12}
\contentsline {subsubsection}{\tocsubsubsection {}{1.2.6}{Twisted curves as root stacks: the markings}}{13}
\contentsline {subsubsection}{\tocsubsubsection {}{1.2.7}{Triviality of relative automorphisms}}{13}
\contentsline {subsubsection}{\tocsubsubsection {}{1.2.8}{Inertia of root stacks when $X$ is a scheme}}{13}
\contentsline {subsubsection}{\tocsubsubsection {}{1.2.9}{Inertia of root stacks when $X$ is an orbifold}}{14}
\contentsline {subsection}{\tocsubsection {}{1.3}{Maps and lifts}}{15}
\contentsline {subsection}{\tocsubsection {}{1.4}{Twisting along a nodal divisor}}{16}
\contentsline {subsubsection}{\tocsubsubsection {}{1.4.3}{Gluing as pushout}}{16}
\contentsline {subsubsection}{\tocsubsubsection {}{1.4.5}{Automorphisms}}{17}
\contentsline {subsubsection}{\tocsubsubsection {}{1.4.8}{}}{18}
\contentsline {subsubsection}{\tocsubsubsection {}{1.4.9}{Twisted curves and twisted nodes}}{18}
\contentsline {subsubsection}{\tocsubsubsection {}{1.4.10}{Maps and lifts}}{18}
\contentsline {section}{\tocsection {}{2}{Expanded pairs and degenerations}}{19}
\contentsline {subsection}{\tocsubsection {}{2.1}{Expanded pairs}}{19}
\contentsline {subsubsection}{\tocsubsubsection {}{2.1.4}{Stacks of expanded pairs}}{20}
\contentsline {subsubsection}{\tocsubsubsection {}{2.1.7}{Defining the stack of expansions in general}}{21}
\contentsline {subsubsection}{\tocsubsubsection {}{2.1.8}{Charts for the stacks}}{22}
\contentsline {subsection}{\tocsubsection {}{2.2}{Twisted expanded degenerations}}{23}
\contentsline {subsubsection}{\tocsubsubsection {}{2.2.5}{Definition in general and charts}}{24}
\contentsline {subsection}{\tocsubsection {}{2.3}{The singular case in the absence of smoothing}}{25}
\contentsline {subsection}{\tocsubsection {}{2.4}{Split expansions}}{26}
\contentsline {subsection}{\tocsubsection {}{2.5}{Weighted and ${\mathfrak {r}}$-twisted expansions}}{27}
\contentsline {subsubsection}{\tocsubsubsection {}{2.5.1}{${\boldsymbol {\Delta }}$-weighted expansions}}{27}
\contentsline {subsubsection}{\tocsubsubsection {}{2.5.2}{${\mathfrak {r}}$-twisted expansions}}{28}
\contentsline {subsection}{\tocsubsection {}{2.6}{Stable expanded configurations}}{29}
\contentsline {section}{\tocsection {}{3}{Stacks of twisted stable maps and their properness}}{30}
\contentsline {subsection}{\tocsubsection {}{3.1}{Conventions on stacks of maps}}{30}
\contentsline {subsection}{\tocsubsection {}{3.2}{Transversal maps and predeformable maps}}{32}
\contentsline {subsubsection}{\tocsubsubsection {}{3.2.5}{Contact orders and Li's predeformability}}{33}
\contentsline {subsection}{\tocsubsection {}{3.3}{Properness of the stack of predeformable maps}}{36}
\contentsline {subsection}{\tocsubsection {}{3.4}{${\mathfrak {r}}$--twisted stable maps and their properness}}{40}
\contentsline {section}{\tocsection {}{4}{Relative and degenerate Gromov-Witten invariants}}{42}
\contentsline {subsection}{\tocsubsection {}{4.1}{Curve classes and orbifold cohomology on nodal DM stacks}}{42}
\contentsline {subsection}{\tocsubsection {}{4.2}{Gromov--Witten invariants for nodal DM stacks}}{42}
\contentsline {subsection}{\tocsubsection {}{4.3}{Relative Gromov--Witten invariants}}{43}
\contentsline {subsection}{\tocsubsection {}{4.4}{Independence of twisting choice}}{44}
\contentsline {subsection}{\tocsubsection {}{4.5}{Invariance under twisting}}{46}
\contentsline {subsection}{\tocsubsection {}{4.6}{Deformation invariance}}{46}
\contentsline {subsection}{\tocsubsection {}{4.7}{Comparison with Jun Li's invariants}}{46}
\contentsline {subsubsection}{\tocsubsubsection {}{4.7.2}{}}{47}
\contentsline {subsubsection}{\tocsubsubsection {}{4.7.3}{}}{47}
\contentsline {subsubsection}{\tocsubsubsection {}{4.7.4}{}}{47}
\contentsline {subsubsection}{\tocsubsubsection {}{4.7.5}{}}{47}
\contentsline {subsection}{\tocsubsection {}{4.8}{Disconnected maps and invariants}}{47}
\contentsline {subsubsection}{\tocsubsubsection {}{4.8.2}{Moduli stacks and their properties}}{48}
\contentsline {subsubsection}{\tocsubsubsection {}{4.8.3}{Disconnected Gromov--Witten invariants}}{48}
\contentsline {subsubsection}{\tocsubsubsection {}{4.8.4}{Contraction morphisms}}{48}
\contentsline {subsubsection}{\tocsubsubsection {}{4.8.7}{An auxiliary stack of expansions}}{49}
\contentsline {section}{\tocsection {}{5}{Degeneration formula}}{51}
\contentsline {subsection}{\tocsubsection {}{5.1}{Setup}}{51}
\contentsline {subsection}{\tocsubsection {}{5.2}{Statement of the formula}}{53}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3}{The Chen--Ruan pairing and Theorem 0.4.1\hbox {}}}{54}
\contentsline {subsection}{\tocsubsection {}{5.3}{Outline of proof of Theorem 5.2.1\hbox {}}}{55}
\contentsline {subsection}{\tocsubsection {}{5.4}{Splitting the coarse target}}{56}
\contentsline {subsection}{\tocsubsection {}{5.5}{Splitting the stack target}}{56}
\contentsline {subsection}{\tocsubsection {}{5.6}{Decomposing the moduli space with split target}}{57}
\contentsline {subsection}{\tocsubsection {}{5.7}{Gluing the target}}{58}
\contentsline {subsection}{\tocsubsection {}{5.8}{Gluing the source}}{59}
\contentsline {subsection}{\tocsubsection {}{5.9}{Comparison with $\Delta $}}{62}
\contentsline {subsection}{\tocsubsection {}{5.10}{End of proof}}{63}
\contentsline {section}{\tocsection {Appendix}{A}{Pairs and nodes}}{64}
\contentsline {subsection}{\tocsubsection {}{A.1}{Smooth and locally smooth pairs}}{64}
\contentsline {subsection}{\tocsubsection {}{A.2}{Transversality for nodal singularities and pairs}}{65}
\contentsline {subsection}{\tocsubsection {}{A.3}{First-order smoothability of nodal singularities}}{66}
\contentsline {section}{\tocsection {Appendix}{B}{Stack constructions}}{66}
\contentsline {subsection}{\tocsubsection {}{B.1}{Using $2$-stacks to define stacks}}{66}
\contentsline {subsection}{\tocsubsection {}{B.2}{Constructing moduli over a base stack}}{67}
\contentsline {subsection}{\tocsubsection {}{B.3}{Inertia stacks of various flavors}}{68}
\contentsline {subsubsection}{\tocsubsubsection {}{B.3.1}{The inertia stack}}{68}
\contentsline {subsubsection}{\tocsubsubsection {}{B.3.2}{Inertia of Deligne--Mumford stacks}}{69}
\contentsline {subsubsection}{\tocsubsubsection {}{B.3.3}{Rigidified inertia}}{69}
\contentsline {subsubsection}{\tocsubsubsection {}{B.3.4}{Cyclotomic inertia and rigidified inertia}}{69}
\contentsline {subsection}{\tocsubsection {}{B.4}{Deformations and obstructions for Artin stacks}}{69}
\contentsline {section}{\tocsection {Appendix}{C}{Stacks of maps and their obstruction theory}}{70}
\contentsline {subsection}{\tocsubsection {}{C.1}{Stacks of maps}}{70}
\contentsline {subsubsection}{\tocsubsubsection {}{C.1.7}{Base change}}{71}
\contentsline {subsubsection}{\tocsubsubsection {}{C.1.8}{Change of family}}{72}
\contentsline {subsection}{\tocsubsection {}{C.2}{Obstruction theory on stacks of maps}}{72}
\contentsline {subsection}{\tocsubsection {}{C.3}{Perfect amplitude}}{74}
\contentsline {section}{\tocsection {Appendix}{}{References}}{75}
\contentsline {section}{\tocsection {Appendix}{D}{Things to do}}{78}