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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.
Keywords
stable maps, quasimaps, elliptic singularities, logarithmic
geometry
Mathematical Subject Classification 2010
Primary: 14N35
Secondary: 14D23
Publication
Received: 1 September 2017
Revised: 5 December 2018
Accepted: 24 March 2019
Published: 30 December 2019
Proposed: Jim Bryan
Seconded: Lothar Göttsche, Dan Abramovich