The moduli space of holomorphic maps from Riemann surfaces to the Grassmannian is known to have two kinds of compactifications: Kontsevich’s stable map compactification and Marian–Oprea–Pandharipande’s stable quotient compactification. Over a non-singular curve, the latter moduli space is Grothendieck’s Quot scheme. In this paper, we give the notion of ‘ ϵ-stable quotients’ for a positive real number ϵ, and show that stable maps and stable quotients are related by wall-crossing phenomena. We will also discuss Gromov–Witten type invariants associated to ϵ-stable quotients, and investigate them under wall crossing.

The Gromov–Witten theory of Deligne–Mumford stacks is a recent development, and hardly any computations have been done beyond three-point genus 0 invariants. This paper provides explicit recursions which, together with some invariants computed by hand, determine all genus 0 invariants of the stack $\mathbb{P}^2_{D,2}$. Here $D$ is a smooth plane curve and $\mathbb{P}^2_{D,2}$ is locally isomorphic to the stack quotient $[U/(\mathbb{Z}/(2))]$, where $U\to V\subseteq \mathbb{P}^2$ is a double cover branched along $D\cap V$. The introduction discusses an enumerative application of these invariants.

In this paper, using the gluing formula of Gromov–Witten invariants for symplectic cutting developed by Li and Ruan, we established some relations between Gromov–Witten invariants of a semipositive symplectic manifold M and its blow-ups along a smooth surface.