On the tautological ring of Humbert curves

TL;DR

This paper studies Humbert curves, a family of genus 5 non-hyperelliptic curves, showing their tautological ring injects into cohomology, and they possess a multiplicative Chow-Künneth decomposition with torsion Ceresa cycle.

Contribution

It demonstrates that Humbert curves have a multiplicative Chow-Künneth decomposition and that their tautological ring injects into cohomology, revealing new structural properties.

Findings

Tautological ring injects into cohomology for Humbert curves

Humbert curves have a multiplicative Chow-Künneth decomposition

Ceresa cycle of Humbert curves is torsion

Abstract

We exhibit a 2-dimensional family of non-hyperelliptic curves of genus 5, called Humbert curves, for which the tautological ring injects into cohomology. In particular, Humbert curves have a multiplicative Chow-K\"unneth decomposition (in the sense of Shen-Vial), and their Ceresa cycle is torsion.