Vol. 13, No. 8, 2019

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

Dhruv Ranganathan, Keli Santos-Parker and Jonathan Wise

Vol. 13 (2019), No. 8, 1765–1805
DOI: 10.2140/ant.2019.13.1765
Abstract

This is the second in a pair of papers developing a framework to apply logarithmic methods in the study of stable maps and singular curves of genus 1. This volume focuses on logarithmic Gromov–Witten theory and tropical geometry. We construct a logarithmically nonsingular and proper moduli space of genus 1 curves mapping to any toric variety. The space is a birational modification of the principal component of the Abramovich–Chen–Gross–Siebert space of logarithmic stable maps and produces logarithmic analogues of Vakil and Zinger’s genus one reduced Gromov–Witten theory. We describe the nonarchimedean analytic skeleton of this moduli space and, as a consequence, obtain a full resolution to the tropical realizability problem in genus 1.

Keywords
logarithmic Gromov–Witten theory, tropical realizability, well spacedness condition
Mathematical Subject Classification 2010
Primary: 14N35
Secondary: 14T05
Milestones
Received: 1 September 2017
Revised: 11 May 2019
Accepted: 4 July 2019
Published: 9 October 2019
Authors
Dhruv Ranganathan
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
United Kingdom
Keli Santos-Parker
Medical School
University of Michigan
Ann Arbor, MI
United States
Jonathan Wise
Department of Mathematics
University of Colorado
Boulder, CO
United States