Let $\pi:\mathcal{C} \to \Delta$ be a family of projective curves of genus $g \ge 2$ over the unit disc $\Delta$, smooth over the punctured disc $\Delta\backslash \{0\}$ and central fiber $\pi^{-1}(0)$ is an irreducible nodal curve with exactly one node. For $t$ close to $0$, denote by $\delta_t$ the vanishing cycle on $H_1(\mathcal{C}_t,\mathbb{Z})$, where $\mathcal{C}_t:=\pi^{-1}(t)$. I am looking for conditions on the central fiber $\pi^{-1}(0)$ such that there exists a $1$-cycle $\gamma \in H_1(\mathcal{C}_t,\mathbb{Z})$ such that $\gamma.\delta_t=1$. Any idea/reference will be most welcome.

If I understand correctly, this is true if $\pi$ is a degeneration of elliptic curves.

Let $\pi:\mathcal{C} \to \Delta$ be a family of projective curves of genus $g \ge 2$ over the unit disc $\Delta$, smooth over the punctured disc $\Delta\backslash \{0\}$ and central fiber $\pi^{-1}(0)$ is an irreducible nodal curve with exactly one node. For $t$ close to $0$, denote by $\delta_t$ the vanishing cycle on $H_1(\mathcal{C}_t,\mathbb{Z})$, where $\mathcal{C}_t:=\pi^{-1}(t)$. I am looking for conditions on the central fiber $\pi^{-1}(0)$ such that there exists a $1$-cycle $\gamma \in H_1(\mathcal{C}_t,\mathbb{Z})$ satisfying: $\gamma.\delta_t=1$. Any idea/reference will be most welcome.

If I understand correctly, this is true if $\pi$ is a degeneration of elliptic curves.