Deformations of Hypersurfaces with Nonconstant Alexander Polynomial

Let X is an element of P-n be an irreducible hypersurface of degree d >= 3 with only isolated semiweighted homogeneous singularities, such that exp(2 pi i/k) is a zero of its 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 >= 3. Moreover, we argue that many of the hypersurfaces with nontrivial Alexander polynomial are limits of constructions of hypersurfaces with not T-smooth deformation spaces. In many instances, this description can be used to find candidates for Alexander-equivalent Zariski pairs.