Universal tropical structures for curves in exploded manifolds

Abstract

For any stable curve $f$ in an exploded manifold, this paper constructs a family of curves $\hat{f}$ with universal tropical structure which contains $f$. Such a family has the property that any other family of curves containing $f$ is locally a small modification of a family which factors through $\hat{f}$. As such, families of curves with universal tropical structure play an important role in the analysis of the moduli stack of curves and the construction of Gromov-Witten invariants of exploded manifolds.

32 pages. Systematizes arguments that first appeared in arXiv:1102.0158. v2: Improved exposition, with typos fixed