Splittability of stellar singular fiber with three branches (Q854367)

Consider a family \(\pi _t:M_t\rightarrow \Delta\) of degenerations of closed Riemann surfaces such that \(\pi _0^{-1}(0)\) is the only normally minimal singular fibre of \(\pi _0\) and for \(t\neq 0\), \(\pi _t\) has more than one singular fibre which are not obtained as blowing ups of smooth fibres. Such germs \(\pi _0\) are called splittable. A singular fibre which is not splittable is called atomic. A singular fibre is called stellar if it has a core and some branches. The authors show that if the number of branches is exactly \(3\), then the degeneration is not atomic.