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

Abstract Let $$\sigma : X\rightarrow \Delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> <mml:mi>Δ</mml:mi> </mml:mrow> </mml:math> 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 $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> 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.