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).

we have a nonsingular cover V 0 → U where V 0 is defined by u 1 v 1 = t. The purity lemma applies to V 0 , so the composition V 0K → C K → M extends over all of V 0. There is a minimal intermediate cover V 0 → V → U such that the family extends already over V ; this V will be of the form xy = t r/m , and the map V → U is given by u = x m , v = y m. Furthermore, there is an action of the group µ m of roots of 1, under which α ∈ µ m sends x to αx and y to α −1 y, and V/µ m = U. This gives the orbispace structure C over C, and the map C K → M extends to a map C → M. This gives the flavor of our definition. We define a category K g,n (M, d), fibered over Sch/S, of twisted stable n-pointed maps C → M of genus g and degree d. This category is given in two equivalent realizations: one as a category of stable twisted M-valued objects over nodal pointed curves endowed with atlases of orbispace charts (see Definition 3.7.2); the other as a category of representable maps from pointed nodal Deligne-Mumford stacks into M, such that the map on coarse moduli spaces is stable (see Definition 4.3.1). In our treatment, both realizations are used in proving our main theorem: Theorem 1.4.1. (1) The category K g,n (M, d) is a proper algebraic stack. (2) The coarse moduli space K g,n (M, d) of K g,n (M, d) is projective. (3) There is a commutative diagram K g,n (M, d) → K g,n (M, d) ↓ ↓ K g,n (M, d) → K g,n (M, d) where the top arrow is proper, quasifinite, relatively of Deligne-Mumford type and tame, and the bottom arrow is finite. In particular, if K g,n (M, d) is a Deligne-Mumford stack, then so is K g,n (M, d).