On the syzygies and Alexander polynomials of nodal hypersurfaces

TL;DR

This paper establishes sharp lower bounds for syzygies of derivatives of polynomials defining nodal hypersurfaces, linking singularity positions to algebraic invariants and exploring implications for Alexander polynomials.

Contribution

It provides new bounds on syzygies of nodal hypersurfaces and connects these bounds to singularity distribution and Alexander polynomial properties.

Findings

Sharp lower bounds for syzygies involving derivatives.

Relation between singularity position and algebraic invariants.

Existence of infinite families with nontrivial Alexander polynomials.

Abstract

We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining a nodal hypersurface. The result gives information on the position of the singularities of a nodal hypersurface expressed in terms of defects or superabundances. The case of Chebyshev hypersurfaces is considered as a test for this result and leads to a potentially infinite family of nodal hypersurfaces having nontrivial Alexander polynomials.