  We introduce the notion of a logarithmic stable map from a minimal log
prestable curve to a log twisted semi-stable variety of form $xy=0$. We study
the compactification of the moduli spaces of such maps and provide a perfect
obstruction theory, applicable to the moduli spaces of (un)ramified stable maps
and stable relative maps. As an application, we obtain a modular
desingularization of the main component of Kontsevich's moduli space of
elliptic stable maps to a projective space.
