Let $Q_t=(xy-z^2-tw^2)\subset\mathbb{P}^3_{x:y:z:w}$ be a family of quadric surfaces with the central fiber $Q_0$ being singular.
Let $C=(xy-z^2,w)\subset Q_t$ be the conic at infinity. For $t\neq 0$ we have $C$ is non-toric in $Q_t$.
But $C$ is toric in $Q_0$?