Polynomial families of tautological classes on $\mathcal{M}_{g,n}^{rt}$

TL;DR

This paper introduces polynomial families of tautological classes on the moduli space of stable genus g curves with rational tails, derived from relative stable maps and their relations to universal Jacobian sections.

Contribution

It defines new classes $P_{g,T}(eta;eta)$ on $ar{ ext{M}}_{g,n}^{rt}$, showing they are polynomial in ramification data and providing methods to compute their coefficients.

Findings

Classes $P_{g,T}(eta;eta)$ are polynomial in ramification data.

Virtual localization yields relations to compute polynomial coefficients.

Comparison with classes $Q_{g,T}$ from universal Jacobian sections.

Abstract

We study classes $P_{g,T}(\alpha;\beta)$ on the moduli space of stable, genus g curves with rational tails defined by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized projective line. A comparison with classes $Q_{g,T}$ arising from sections of the universal Jacobian shows the classes $P_{g,T}(\alpha;\beta)$ are polynomial in the parts of the partitions indexing the special ramification data. Virtual localization on moduli spaces of relative stable maps gives sufficient relations to compute the coefficients of these polynomials in various cases.