Introduction 1 2. Chow rings, cohomology and homology of stacks 5 3. The cyclotomic inertia stack and its rigidification 10 4. Twisted curves and their maps 18 5. The boundary of moduli 22 6. Gromov-Witten classes 27 7. The age grading 35 8. A few useful facts 41 9. An example: the weighted projective line 44 Appendix A. Gluing of algebraic stacks along closed substacks 48 Appendix B. Taking roots of line bundles 52 Appendix C. Rigidification 54 References 57 Research of D.A.

Contents 1. Introduction 1 2. Terminology 6 3. Deformation theory of twisted covers 9 4. Twisted covers and admissible covers 11 5. Rigidification and Teichmüller structures 14 6. Abelian twisted level structures 19 7. Automorphisms of twisted G-covers 22 Appendix A. Some remarks on étale cohomology of Deligne-Mumford stacks 33 References 35 [W] S. Wewers, Construction of Hurwitz spaces, Institut für Experimentelle Mathematik preprint No. 21 (1998).

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