Moduli of stable maps in genus one and logarithmic geometry, I

Abstract

This is the first in a pair of papers developing a framework for the application of logarithmic structures in the study of singular curves of genus 1.We construct a smooth and proper moduli space dominating the main component of Kontsevich's space of stable genus 1 maps to projective space.A variation on this theme furnishes a modular interpretation for Vakil and Zinger's famous desingularization of the Kontsevich space of maps in genus 1.Our methods also lead to smooth and proper moduli spaces of pointed genus 1 quasimaps to projective space.Finally, we present an application to the log minimal model program for M 1;n .We construct explicit factorizations of the rational maps among Smyth's modular compactifications of pointed elliptic curves.14N35; 14D23 1.