Some consequences of the $μ$-constant condition for families of surfaces

Let $f : X\to Δ$ be a $1$-parameter family of $2$-dimensional isolated hypersurface singularities. In this paper, we show that if the Milnor number is constant, then any semistable model, obtained from $f$ after a sufficiently large base change must satisfy non trivial restrictions. Those restrictions are in terms of the dual complex, Hodge structure and numerical invariants of the central fibre.