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

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

doi:10.2140/ant.2019.13.1765

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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

Vol. 13, No. 8, 2019