Generic Manin-Mumford

Given a collection of algebraic numbers $\mathcal{S}\subset \overline{\mathbb{Q}}$ we study the varieties $V$ in $\mathbb{A}^m_\mathbb{C}$ such that $V(\mathbb{C})\cap\mathcal{S}^m$ is Zariski-dense in $V$. We show that for many classical families of algebraic numbers $\mathcal{S}$---such as the fam