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 ${\mathcal{M}}_g^{\mathrm{a}\mathrm{m}\mathrm{p}}$ 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 ${\mathcal{M}}_g^{\mathrm{a}\mathrm{m}\mathrm{p}}$ have a lower bound and satisfy the Northcott property.