Heights of Ceresa and Gross-Schoen cycles

We study the Beilinson-Bloch heights of Ceresa and Gross-Schoen cycles in families. We construct that for any $g\ge 3$, a Zariski open dense subset $\mathcal{M}_g^{\mathrm{amp}}$ of $\mathcal{M}_g$, the coarse moduli of curves of genus $g$ over $\mathbb{Q}$, such that the heights of Ceresa cycles and Gross-Schoen cycles over $\mathcal{M}_g^{\mathrm{amp}}$ have a lower bound and satisfy the Northcott property.

💡 Research Summary

This paper investigates the Beilinson‑Bloch heights of two distinguished families of homologically trivial cycles on curves: the Ceresa cycles and the Gross‑Schoen cycles. For every genus g ≥ 3 the authors construct a Zariski‑open dense subset 𝑀_g^{amp} of the coarse moduli space 𝑀_g of smooth projective curves over ℚ. Over this “ample locus” they prove two fundamental properties of the heights:

1. Uniform lower bound – there exist positive constants ε(g) and c(g) such that for every rational point s in the pre‑image π⁻¹(𝑈)(ℚ) (the universal Picard of degree 1 over 𝑀_g) the Beilinson‑Bloch height of the Gross‑Schoen cycle satisfies
   \