Algebraic Geometry Seminar: A smooth space of stable maps and a conjecture of Abramovich--Fantechi

The stack of stable maps parameterizes maps from a complete curves having at worst nodal singularities into a smooth scheme.  Generally this stack is not smooth, but we will explain how it can be made smooth by relaxing the condition that the source curves be complete.  Although the resulting stack is not fibered in groupoids, and therefore may not be easily accessible to geometric intuition, it is a natural setting in which to construct the virtual fundamental class.  We will discuss how this generalization can be used to prove a conjecture of Abramovich and Fantechi relating the virtual fundamental classes of two different moduli spaces parameterizing stable maps into mildly singular schemes.