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

Ranganathan, Dhruv; Santos-Parker, Keli; Wise, Jonathan

doi:10.2140/gt.2019.23.3315

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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

DOI: 10.2140/gt.2019.23.3315

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 ${\bar{ℳ}}_{1, n}$ . We construct explicit factorizations of the rational maps among Smyth’s modular compactifications of pointed elliptic curves.

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Keywords

stable maps, quasimaps, elliptic singularities, logarithmic geometry

Mathematical Subject Classification 2010

Primary: 14N35

Secondary: 14D23

References

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

Authors

Dhruv Ranganathan
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge United Kingdom

Keli Santos-Parker
Department of Mathematics University of Colorado Boulder, CO United States	Medical School University of Michigan Ann Arbor, MI United States

Jonathan Wise
Department of Mathematics University of Colorado Boulder, CO United States

Volume 23, issue 7 (2019)