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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

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 ̄1,n. We construct explicit factorizations of the rational maps among Smyth’s modular compactifications of pointed elliptic curves.

stable maps, quasimaps, elliptic singularities, logarithmic geometry
Mathematical Subject Classification 2010
Primary: 14N35
Secondary: 14D23
Received: 1 September 2017
Revised: 5 December 2018
Accepted: 24 March 2019
Published: 30 December 2019
Proposed: Jim Bryan
Seconded: Lothar Göttsche, Dan Abramovich
Dhruv Ranganathan
Department of Pure Mathematics and Mathematical Statistics
University of 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