Line-Bundles on Stacks of Relative Maps

TL;DR

This paper develops technical tools involving line-bundles on moduli stacks of relative maps, crucial for defining and computing relative Gromov-Witten invariants, including degeneration formulas and the Trivial Cylinders Theorem.

Contribution

It introduces new relations between line-bundles on moduli stacks and proves the Trivial Cylinders Theorem, advancing the formalism for relative Gromov-Witten invariants.

Findings

Relations between line-bundles yield degeneration formulas

Proof of the Trivial Cylinders Theorem

Enhanced formalism for relative Gromov-Witten invariants

Abstract

This paper provides some technical results needed in "Formalism for Relative Gromov-Witten Invariants." We study line-bundles on the moduli stacks of relative stable and rubber maps that are used to define relative Gromov-Witten invariants in the sense of J. Li. Relations between these line-bundles yield degeneration formulae. In addition we prove a technical result called the Trivial Cylinders Theorem.