Hypersurfaces with many Aj-singularities: explicit constructions

TL;DR

This paper presents explicit constructions of algebraic hypersurfaces with numerous Aj-singularities using polynomials derived from line arrangements, generalizing folding polynomials and enhancing singularity configurations.

Contribution

It introduces new methods to construct hypersurfaces with many Aj-singularities based on arrangements of lines and associated polynomials, expanding the class of known singularity-rich surfaces.

Findings

Construction of hypersurfaces with many Aj-singularities

Use of polynomials from line arrangements including folding polynomials

Explicit examples demonstrating the abundance of singularities

Abstract

A construction of algebraic surfaces based on two types of simple arrangements of lines, containing the prototiles of substitution tilings, has been proposed recently. The surfaces are derived with the help of polynomials obtained from products of the lines generating the simple arrangements. One of the arrangements gives the generalizations of the Chebyshev polynomials known as folding polynomials. The other produces a family of polynomials having more critical points with the same critical values, which can also be used to derive hypersurfaces with many Aj-singularities.