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 projective space. The new approach provides complete local structural results for this moduli space as well as for the desingularization of the entire moduli space and should fully extend to higher genera. 
Yi Hu and Jun Li