We define a Deligne-Mumford stack XD,r which depends on a scheme X, an effective Cartier divisor D ⊂ X, and a positive integer r. Then we show that the Abramovich-Vistoli moduli stack of stable maps into XD,r provides compactifications of the locally closed substacks of M g,n(X, β) corresponding to relative stable maps.

We define Donaldson-Thomas invariants of Calabi-Yau orbifolds and we develop a topological vertex formalism for computing them. The basic combinatorial object is the orbifold vertex V G λµν , a generating function for the number of 3D partitions asymptotic to 2D partitions λ, µ, ν and colored by representations of a finite Abelian group G acting on C 3 . In the case where G ∼ = Z n acting on C 3 with transverse A n−1 quotient singularities, we give an explicit formula for V G λµν in terms of Schur functions. We discuss applications of our formalism to the Donaldson-Thomas Crepant Resolution Conjecture and to the orbifold Donaldson-Thomas/Gromov-Witten correspondence. We also explicitly compute the Donaldson-Thomas partition function for some simple orbifold geometries: the local football P 1 a,b and the local BZ 2 gerbe. arXiv:1008.4205v1 [math.AG] 25 Aug 2010 42 7.3. n-quotient, n-core, and the retrograde 48 Appendix A. Grothendieck-Riemann-Roch for orbifolds and the Toen operator. 60 Appendix B. Orbifold toric CY3s and web diagrams 63 B.1. Reading off the local model at a point from the web diagram 65 B.2. Reading off the local data at a curve from the web diagram 66 References 68

Since Jun Li's original definition, several other definitions of expanded pairs and expanded degenerations have appeared in the literature. We explain how these definitions are related and introduce several new variants and perspectives. Among these are the twisted expansions used by Abramovich and Fantechi as a basis for orbifold techniques in degeneation formulas.

We prove that genus 0 Gromov-Witten invariants of a smooth scheme relative to a smooth divisor coincide with genus 0 orbifold Gromov-Witten invariants of an appropriate root stack construction along the divisor. The proof is given at the level of virtual fundamental classes.

We use twisted stable maps to compute the number of rational degree d plane curves having prescribed contacts to a smooth plane cubic.

We give a construction of the moduli space of stable maps to the classifying stack Bµ r of a cyclic group by a sequence of rth root constructions on M 0,n . We prove a closed formula for the total Chern class of µ r -eigenspaces of the Hodge bundle, and thus of the obstruction bundle of the genus-zero Gromov-Witten theory of stacks of the form [C N /µ r ]. We deduce linear recursions for genus-zero Gromov-Witten invariants.

Abstract. We use three different methods to count the number of lines in the plane whose intersection with a fixed general quintic has fixed cross-ratios. We compare and contrast these methods, shedding light on some classical ideas which underly modern techniques.

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