Deformations of Hypersurfaces with Nonconstant Alexander Polynomial

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