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We describe an algebro-geometric approach to Vakil-Zinger's desingularizationof the main component of the moduli of genus one stable maps to projectivespace. The new approach provides complete local structural results for thismoduli space as well as for the desingularization of the entire moduli spaceand should fully extend to higher genera.

Genus-One Stable Maps, Local Equations, and Vakil-Zinger's desingularization