The boundary of the Milnor fiber of Hirzebruch surface singularities

Abstract

We give the first (as far as we know) complete description of the boundary of the Milnor fiber for some non-isolated singular germs of surfaces in ${\bf C}^3$. We study irreducible (i.e. $gcd (m,k,l) = 1$) non-isolated (i.e. $1 \leq k < l$) Hirzebruch hypersurface singularities in ${\bf C}^3$ given by the equation $z^m - x^ky^l = 0$. We show that the boundary $L$ of the Milnor fiber is always a Seifert manifold and we give an explicit description of the Seifert structure. From it, we deduce that : 1) $L$ is never diffeomorphic to the boundary of the normalization. 2) $L$ is a lens space iff $m = 2$ and $k = 1$. 3) When $L$ is not a lens space, it is never orientation preserving diffeomorphic to the boundary of a normal surface singularity.