COMPARISON OF TWO DESINGULARIZATIONS OF THE MODULI SPACE OF ELLIPTIC STABLE MAPS

Abstract

. We study the geometry of the moduli space of elliptic stable maps to projective space. The main component of the moduli space of elliptic stable maps is singular. There are two different ways to desingu-larize this space. One is Vakil-Zinger’s desingularization and the other is via the moduli space of logarithmic stable maps. Our main result is a proof of the direct geometric relationship between these two spaces. For degree less than or equal to 3, we prove that the moduli space of logarithmic stable maps can be obtained by blowing up Vakil-Zinger’s desingularization.