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

Jorge Martín-Morales, Wayne Ng Kwing King

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.

Modules of logarithmic derivations in weighted projective spaces and applications to 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.

Paper Structure

This paper contains 15 sections, 27 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Module of logarithmic derivations
  4. Eigenscheme and Multiple eigenscheme
  5. Module of logarithmic derivations for weighted projective spaces
  6. Weighted projective space
  7. Module of logarithmic derivations for $\mathbb{P}^n_\omega$
  8. Weighted Eigenschemes
  9. From free divisors in $\mathbb{P}_\omega^{n-1}$ to free divisors in $\mathbb{A}^{n}$ and the Cone Construction for $\mathbb{P}^{n}$
  10. Cone Construction for $\mathbb{P}^{n}$
  11. Homogenization and Freeness in projective space
  12. Applications: New free divisors in affine and projective spaces
  13. New free divisors coming from complete reflection arrangements
  14. New free divisors in $\mathbb{P}^3$, a variant of Brieskorn-Pham polynomials example in Buchweitz_Conca_2012
  15. New free divisors coming from pencils of hypersurfaces example in digennaro2025saitostheoremrevisitedapplication

Key Result

Lemma 3.2

(See e.g. Iano-Fletcher_2000)

Theorems & Definitions (74)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • ...and 64 more