Moduli of stable maps in genus one and logarithmic geometry I

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

37 pages, 4 figures. Final version to appear in Geometry & Topology

References in corpus (1)

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

Cited by in corpus (13)

• A moduli stack of tropical curves
• Gromov-Witten theory with maximal contacts
• Logarithmic compactification of the Abel-Jacobi section
• 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
• A classification of modular compactifications of the space of pointed elliptic curves by Gorenstein curves
• Tangent curves to degenerating hypersurfaces
• Modular compactifications of ${\mathcal{M}}_{2,n}$ with Gorenstein singularities
• Hyperelliptic Gorenstein curves and logarithmic differentials
• Gorenstein curve singularities of genus three
• A geographical study of ${\overline{\mathcal{M}}}_2({\mathbb{P}}^2,4{)}^{\text{main}}$
• The dual complex of ${\mathcal{M}}_{1,n}({\mathbb{P}}^r,d)$ via the geometry of the Vakil--Zinger moduli space
• Vanishing of Brauer groups of moduli stacks of stable curves