Logarithmic moduli of roots of line bundles on curves

TL;DR

This paper develops a framework using logarithmic line bundles to construct compactified moduli spaces of roots of line bundles on curves, incorporating tropical and logarithmic Jacobians, and provides a tautological formula for the double ramification cycle.

Contribution

It introduces a new approach to compactify spaces of roots on curves using logarithmic line bundles and tropical geometry, extending previous work and deriving a tautological formula for the double ramification cycle.

Findings

Constructed compactified moduli spaces of roots of line bundles.

Connected the theory to tropical and logarithmic Jacobians.

Derived a tautological formula for the double ramification cycle.

Abstract

We use the theory of logarithmic line bundles to construct compactifications of spaces of roots of a line bundle on a family of curves, generalising work of a number of authors. This runs via a study of the torsion in the tropical and logarithmic jacobians (recently constructed by Molcho and Wise). Our moduli space carries a `double ramification cycle' measuring the locus where the given root is isomorphic to the trivial bundle, and we give a tautological formula for this class in the language of piecewise polynomial functions (as recently developed by Molcho-Pandharipande-Schmitt and Holmes-Schwarz).