Stable maps to rational curves and the relative Jacobian

TL;DR

This paper proves an equivalence between two geometric cycles on the moduli space of compact type curves, extending previous results from rational tail curves to a broader setting involving relative stable maps and the universal Jacobian.

Contribution

It establishes a new identity between cycles defined via virtual fundamental classes and the Abel section of the universal Jacobian on the moduli space of curves.

Findings

The two cycles coincide on the moduli space of compact type curves.

Extension of previous results from rational tail curves to more general compact type curves.

Provides new tools for understanding the geometry of the moduli space.

Abstract

We consider two cycles on the moduli space of compact type curves and prove that they coincide. The first is defined by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized rational curve, while the second is obtained as the intersection of the Abel section of the universal Jacobian with the zero section. Our comparison extends results of Cavalieri-Marcus-Wise where the same identity was proved over on the locus of rational tails curves.