Logarithmic stable maps

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.