Moduli spaces of stable quotients and wall-crossing phenomena

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 nonsingular curve, the latter moduli space is Grothendieck's Quot scheme. In this paper, we give the notion of 'epsilon-stable quotients' for a positive real number epsilon, and show that stable maps and stable quotients are related by wall-crossing phenomena. We will also discuss Gromov-Witten type invariants associated to epsilon-stable quotients, and investigate them under wall crossing.