While reading the literature on toric degeneration problems, I notice that sometimes the degeneration problem is described as following:
Whether there exist a flat family $\pi:\mathcal{F} \to \text{Spec}\mathbb{C}[t]$ such that generic fiber $\pi^{-1}(t_1)\cong X_1$ (say) and special fiber $\pi^{-1}(0)\cong X_0$. I am happy with this definition because I think it is geometric because $\text{Spec}\mathbb{C}[t]$ represents the affine line.
However, when people consider specialization of curve $\pi: \mathcal{F}\to \text{Spec}\mathbb{C}[[t]]$, Instead of polynomial ring, power series is used. Is there any geometric reason behind it?