Genus-one stable maps, local equations, and Vakil-Zinger's desingularization

We describe an algebro-geometric approach to Vakil-Zinger's desingularization of the main component of the moduli of genus one stable maps to ℙⁿ (Vakil and Zinger in Res Announc Am Math Soc 13:53-59, 2007; Geom Topol 12(1):1-95, 2008). Our approach is based on understanding the local structure of this moduli space; it also gives a partial desingularization of the entire moduli space. The results proved should extend to higher genera.