Tautological ring of the moduli space of generalised parabolic line bundles on a curve

TL;DR

This paper studies the tautological ring of the normalization of the compactified Jacobian of a nodal curve, showing it is generated by pullbacks of Brill-Noether classes and natural classes, under various algebraic operations.

Contribution

It characterizes the structure of the tautological ring for a specific class of algebraic curves, extending understanding of algebraic classes on moduli spaces.

Findings

The tautological ring is generated by pullbacks of Brill-Noether classes.

The ring is stable under algebraic operations like multiplication, Pontryagin product, and Fourier transform.

The structure is explicitly described for the normalization of a nodal curve.

Abstract

In this paper, we consider the tautological ring containing the extended Brill-Noether algebraic classes on the normalization of the compactified Jacobian of a complex nodal projective curve (with one node). This smallest $\Q$-subalgebra of algebraic classes under algebraic equivalence, stable under extensions of the maps induced by multiplication maps, Pontrayagin product and Fourier transform, is shown to be generated by pullback of the Brill-Noether classes of the Jacobian of the normalized curve and some natural classes.