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?



Degenerations of hyperelliptic coverings
If $p\_1 = p\_2 \ne p\_3 = p\_4 \ne p\_5 \ne p\_6$ then the normalization of the double cover branched at the divisor $D = \sum\_{i=1}^6 p\_i$ is a smooth irreducible rational curve. If also $p\_5$ and $p\_6$ collide, the normalization of the double cover is the union of two smooth rational curves.


