#### Volume 23, issue 7 (2019)

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

### Dhruv Ranganathan, Keli Santos-Parker and Jonathan Wise

Geometry & Topology 23 (2019) 3315–3366
##### 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 ${\stackrel{̄}{\mathsc{ℳ}}}_{1,n}$. We construct explicit factorizations of the rational maps among Smyth’s modular compactifications of pointed elliptic curves.

##### Keywords
stable maps, quasimaps, elliptic singularities, logarithmic geometry
Primary: 14N35
Secondary: 14D23