  Let X be an irreducible hypersurface in $\mathbb{P}^n$ of degree $d\geq 3$
with only isolated semi-weighted homogeneous singularities, such that
$exp(\frac{2\pi i}{k})$ is a zero of the Alexander polynomial. Then we show
that the equianalytic deformation space of $X$ is not $T$-smooth except for a
finite list of triples $(n,d,k)$.
  This result captures the very classical examples by B. Segre of families of
degree $6m$ plane curves with $6m^2$, $7m^2$, $8m^2$ and $9m^2$ cusps, where
$m\geq 3$.
  Moreover, we argue that many of the hypersurfaces with non-trivial Alexander
polynomial are limits of constructions of hypersurfaces with not $T$-smooth
deformation spaces. In many instances this description can be used to construct
Alexander-equivalent Zariski pairs.
