Chern Classes of Logarithmic Derivations for Free Divisors with Jacobian Ideal of Linear Type

TL;DR

This paper computes the Chern class of logarithmic derivations for free divisors with Jacobian ideal of linear type and confirms a related conjecture, linking it to the Chern-Schwartz-MacPherson class of the complement.

Contribution

It provides an explicit computation of Chern classes for a class of free divisors and verifies a conjecture connecting these classes with Chern-Schwartz-MacPherson classes.

Findings

Confirmed Aluffi's conjecture for free divisors with Jacobian ideal of linear type.

Established a relationship between Chern classes of logarithmic derivations and Chern-Schwartz-MacPherson classes.

Extended understanding of characteristic classes in the context of free divisors.

Abstract

Let $X$ be a nonsingular variety defined over an algebraically closed field of characteristic $0$, and $D$ be a free divisor with Jacobian ideal of linear type. We compute the Chern class of the sheaf of logarithmic derivations along $D$ and compare it with the Chern-Schwartz-MacPherson class of the hypersurface complement. Our result establishes a conjecture by Aluffi raised in \cite{hyparr}.