Pullbacks of universal Brill-Noether classes via Abel-Jacobi morphisms

TL;DR

This paper computes the Chern character of certain derived pushforwards on the universal curve and uses this to express pullbacks of Brill-Noether classes via Abel-Jacobi sections to universal Jacobians.

Contribution

It provides a general formula for pullbacks of Brill-Noether classes through Abel-Jacobi morphisms on compactified universal Jacobians.

Findings

Explicit formulas for Chern characters of pushforwards.

Expression of Brill-Noether class pullbacks via Abel-Jacobi sections.

Applicability to various compactifications of universal Jacobians.

Abstract

Following Mumford and Chiodo, we compute the Chern character of the derived pushforward $\textrm{ch} (R^\bullet\pi_\ast\mathscr{O}(\mathsf{D}))$, for $\mathsf D$ an arbitrary element of the Picard group of the universal curve over the moduli stack of stable marked curves. This allows us to express the pullback of universal Brill-Noether classes via Abel-Jacobi sections to the compactified universal Jacobians, for all compactifications such that the section is a well-defined morphism.