On the height of Gross-Schoen cycles in genus three

TL;DR

This paper demonstrates that for genus three curves over rationals, the height of a canonical Gross-Schoen cycle can grow without bound, revealing new insights into their geometric properties.

Contribution

It establishes the existence of a sequence of genus three curves with unbounded Gross-Schoen cycle heights over the rationals.

Findings

Gross-Schoen cycle height tends to infinity for certain genus three curves

Sequence of genus three curves with unbounded cycle heights over rationals

New understanding of geometric complexity in genus three curves

Abstract

We show that there exists a sequence of genus three curves defined over the rationals in which the height of a canonical Gross-Schoen cycle tends to infinity.