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.



Degeneration of smooth curves and Picard-Lefschetz formula
Because the symplectic form on $H\_1(C\_t, \mathbb Z)$ is a perfect pairing, it suffices to check that there is a group homomorphism $H\_1(C\_t,\mathbb Z) \to \mathbb Z$ that sends $ \gamma$ to $1$, which follows if $\gamma$ is not divisible by any $n>1$ in $H\_1(C\_t,\mathbb Z)$.


Because $\gamma$ is defined as the generator of the kernel of $H\_1(C\_t,\mathbb Z) \to H\_1(C\_0,\mathbb Z)$, $\gamma$ is indivisible as long as $H\_1(C\_0,\mathbb Z)$ is nontorsion, which follows from it being an irreducible nodal curve.


