Help me please.

Let $k$ be an algebraically closed field (I am mainly interested in $k = \overline{\mathbb{Q}}, \overline{\mathbb{F}_q}$). Consider a plane curve $C \subset \mathbb{A}^2$ of degree $d$ over the rational function field $k(t)$. Suppose that $C$ is absolutely irreducible, i.e., irreducible over $\overline{k(t)}$.

Am I right that there is only the finite number $n$ of elements $t \in k$ such that the reduction of $C$ to $k$ is reducible ? Is there an upper bound for $n$?

Thank you in advance.