Moduli of stable maps in genus one and logarithmic geometry II

Abstract

This is the second in a pair of papers developing a framework to apply logarithmic methods in the study of singular curves of genus $1$. This volume focuses on logarithmic Gromov--Witten theory and tropical geometry. We construct a logarithmically nonsingular 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 an enumerative genus $1$ curve counting theory. We describe the non-archimedean analytic skeleton of this moduli space and, as a consequence, obtain a full resolution to the tropical realizability problem in genus $1$.

36 pages, 5 figures. Final version to appear in Algebra & Number Theory

Cited by in corpus (9)

• A moduli stack of tropical curves
• Moduli of stable maps in genus one and logarithmic geometry I
• A smooth compactification of the space of genus two curves in projective space via logarithmic geometry and Gorenstein curves
• Curve counting in genus one: elliptic singularities & relative geometry
• Modular compactifications of ${\mathcal{M}}_{2,n}$ with Gorenstein singularities
• Hyperelliptic Gorenstein curves and logarithmic differentials
• Moduli spaces of codimension-one subspaces in a linear variety and their tropicalization
• Gorenstein curve singularities of genus three
• Constructions of superabundant tropical curves in higher genus