Take six distinct points $p_1,\dots,p_6\in\mathbb{P}^1$ and consider the double covering $f:C\rightarrow \mathbb{P}^1$ ramified over $p_1,\dots,p_6\in\mathbb{P}^1$. Then $C$ is a smooth curve of genus two.

Can we degenerate $C$ to a singular rational curve or to a union of smooth rational curves by collapsing some of the $p_i$ together?