Local cohomology of logarithmic forms

TL;DR

This paper studies the geometric and homological properties of divisors on smooth algebraic varieties, focusing on logarithmic forms, free divisors, and hyperplane arrangements, with explicit calculations for generic cases.

Contribution

It provides a detailed analysis of the local cohomology of logarithmic forms and explores the relationship between free divisors and hyperplane arrangements, including explicit computations.

Findings

Complete calculation of local cohomology for generic hyperplane arrangements

Insights into the geometry of Jacobian schemes of divisors

Connections between free divisors and logarithmic forms

Abstract

Let Y be a divisor on a smooth algebraic variety X. We investigate the geometry of the Jacobian scheme of Y, homological invariants derived from logarithmic differential forms along Y, and their relationship with the property that Y is a free divisor. We consider arrangements of hyperplanes as a source of examples and counterexamples. In particular, we make a complete calculation of the local cohomology of logarithmic forms of generic hyperplane arrangements.