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

Abstract

We describe an algebro-geometric approach to Vakil–Zinger’s desingularization of the main component of the moduli of genus one stable maps to $${\mathbb{P}^{n}}$$ (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.