The boundary of the Milnor fiber of Hirzebruch surface singularities

F. Michel, Anne Pichon, C. Weber

Published 2005-09-20Version 1

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.