Compactified universal jacobian and the double ramification cycle

TL;DR

This paper provides new algebraic formulas for the double ramification cycle using the compactified universal jacobian, extending Abel-Jacobi maps and recovering known divisor pullback formulas.

Contribution

It introduces a novel expression for the double ramification cycle in the Chow ring via the zero section of the compactified universal jacobian and extends Abel-Jacobi maps to treelike curves.

Findings

Derived a formula for the double ramification cycle in the Chow ring.

Extended Abel-Jacobi maps to treelike curves.

Reproduced known formulas for theta divisor pullbacks.

Abstract

Using the compactified universal jacobian over the moduli space of stable marked curves, we give an expression in terms of natural classes of the zero section of the compactified universal jacobian the (rational) Chow ring. After extending variants of the Abel-Jacobi map to a locus containing curves of treelike type we give a formula for the pullback of the said zero section along these extensions. The same approach is also applied to recover known formulas for the pullback of theta divisors to the moduli space of marked stable curves.