Koszul complexes and pole order filtrations

TL;DR

This paper explores the relationship between Koszul complex cohomology and pole order filtrations on the cohomology of complements of hypersurfaces, providing tools to compute filtrations and analyze syzygies, especially for nodal hypersurfaces.

Contribution

It introduces spectral sequences linking Koszul complexes and pole order filtrations, enabling the determination of filtrations and syzygy analysis for hypersurfaces, notably nodal ones.

Findings

Spectral sequences relate Koszul cohomology to pole order filtrations.

Methods to compute filtrations for curves and surfaces.

For nodal surfaces in P^3, F^2H^3 differs from P^2H^3 when degree ≥ 4.

Abstract

We study the interplay between the cohomology of the Koszul complex of the partial derivatives of a homogeneous polynomial $f$ and the pole order filtration $P$ on the cohomology of the open set $U=\PP^n \setminus D$, with $D$ the hypersurface defined by $f=0$. The relation is expressed by some spectral sequences, which may be used on one hand to determine the filtration $P$ in many cases for curves and surfaces, and on the other hand to obtain information about the syzygies involving the partial derivatives of the polynomial $f$. The case of a nodal hypersurface $D$ is treated in terms of the defects of linear systems of hypersurfaces of various degrees passing through the nodes of $D$. When $D$ is a nodal surface in $\PP^3$, we show that $F^2H^3(U) \ne P^2H^3(U)$ as soon as the degree of $D$ is at least 4.