Modules of logarithmic derivations in weighted projective spaces and applications to free divisors

TL;DR

This paper generalizes the module of logarithmic derivations to weighted projective spaces, providing criteria and methods to identify and construct large families of free divisors with explicit examples.

Contribution

It introduces a weighted version of the module of logarithmic derivations and generalizes Saito's criterion for freeness in weighted projective spaces.

Findings

Generalized Saito's criterion using weighted multiple eigenschemes

Constructed new families of free divisors in affine and projective spaces

Provided numerous explicit examples of free divisors

Abstract

We introduce a weighted version of the module of logarithmic derivations of a divisor in weighted projective space, and provide a generalization of Saito's criterion for freeness in terms of weighted multiple eigenschemes (wME-schemes). Freeness of the nonstandard Z-graded module allows one to consider big families of free divisors in affine and standard projective space, i.e. when the module of logarithmic derivations of the divisor is free over the respective coordinate rings. We present a method to identify and construct these new families of free divisors in affine and projective space in any dimension, and give numerous explicit examples.