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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.

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  • $\begingroup$ There exist simple closed curves $a,b$ on $C_t$ whose homology classes (also denoted $a,b$) satisfy $a\cdot b = 1$. Since the central fiber is irreducible with exactly one node, the vanishing cycle $\delta_t$ is a non-separating simple closed curve on $C_t$; it follows from the classification of surfaces that there is an orientation-preserving homeomorphism $f$ of $C_t$ such that $f(b) = \delta_t$. You may then take $\gamma = f(a)$. $\endgroup$ – Tony Aug 3 '18 at 14:47
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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.

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