Deformations of hypersurfaces with non-constant Alexander polynomial

TL;DR

This paper investigates how hypersurfaces with specific singularities and non-trivial Alexander polynomials have deformation spaces that are often not smooth, revealing classical examples and constructing Zariski pairs.

Contribution

It characterizes the non-smoothness of equianalytic deformation spaces for hypersurfaces with certain Alexander polynomial zeros, extending classical examples and providing new constructions.

Findings

Most hypersurfaces with non-trivial Alexander polynomial have non-$T$-smooth deformation spaces.

Classical examples by Segre are encompassed within the results.

The paper constructs Alexander-equivalent Zariski pairs using these insights.

Abstract

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.