On the syzygies and Alexander polynomials of nodal hypersurfaces

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.

Hilbert Series and Lefschetz Properties of Dimension One Almost Complete Intersections

“…properties holding for nodal hypersurfaces, characterized by the fact that they are the hypersurfaces with isolated singularities satisfying I = J f ). Such results have been obtained in [8], [9], [10] using rather advanced Hodge theory.…”

Jacobian syzygies, stable reflexive sheaves, and Torelli properties for projective hypersurfaces with isolated singularities

“…It is interesting to note that even though the approaches in [9] and [19] are quite different, the condition that the singularities of V are weighted homogeneous plays a key role in both papers. While this inequality is the best possible in general, as one can see by considering hypersurfaces with a lot of singularities, see [10], [7], for situations when the hypersurface V has a small number of singularities this result is far from optimal. Our first result gives the following better bound in this case.…”

On the syzygies and Hodge theory of nodal hypersurfaces

“…where a j ∈ S r , modulo the trivial, or Koszul, syzygies generated by (f j )f i + (−f i )f j = 0 for all i < j. The following result was proved in [16].…”