Some consequences of the $\mu$-constant condition for families of surfaces

Abstract

Let $f:X\to \Delta$ 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.