Relations between tautological cycles on Jacobians

TL;DR

This paper investigates the structure of tautological cycles on Jacobians, providing new results on their ring, vanishing properties, and explicit relations, with implications for algebraic geometry and cycle theory.

Contribution

It introduces a new theorem on the tautological ring of a general curve, improving dimension calculations and cycle relations, and extends existing results to the Chow ring.

Findings

New theorem on tautological ring structure

Vanishing result for generating classes p_i

Method for explicit cycle relations

Abstract

We study tautological cycle classes on the Jacobian of a curve. We prove a new result about the ring of tautological classes on a general curve that allows, among other things, easy dimension calculations and leads to some general results about the structure of this ring. Next we obtain a vanishing result for some of the generating classes p_i; this gives an improvement of an earlier result of Herbaut. Finally we lift a result of Herbaut and van der Geer-Kouvidakis to the Chow ring (as opposed to its quotient modulo algebraic equivalence) and we give a method to obtain further explicit cycle relations. As an ingredient for this we prove a theorem about how Polishchuk's operator D lifts to the tautological subalgebra of Chow(J).