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arxiv: 2603.26574 · v2 · submitted 2026-03-27 · 🧮 math.AC

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

Jorge Mart\'in-Morales , Wayne Ng Kwing King (LMAP) This is my paper

Pith reviewed 2026-05-14 22:25 UTC · model grok-4.3

classification 🧮 math.AC
keywords logarithmic derivationsfree divisorsweighted projective spaceSaito criterionweighted multiple eigenschemesnonstandard graded moduleshypersurface singularities
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The pith

A weighted version of the logarithmic derivation module is free exactly when its weighted multiple eigenschemes satisfy a Saito-type criterion, producing explicit families of free divisors in affine and projective space of any dimension.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a nonstandard Z-graded module of logarithmic derivations adapted to weighted projective space. It gives a weighted generalization of Saito's freeness criterion expressed through weighted multiple eigenschemes. When the module is free, the corresponding divisor becomes free over the coordinate ring, allowing systematic construction of large families of such divisors in ordinary affine and projective spaces. The authors supply a concrete method to locate and build these families together with many explicit examples in every dimension.

Core claim

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.

What carries the argument

The weighted multiple eigenschemes (wME-schemes) that generalize Saito's criterion to detect freeness of the nonstandard Z-graded logarithmic derivation module.

If this is right

  • Large explicit families of free divisors become available in affine space of arbitrary dimension.
  • The same families exist in standard projective space.
  • Freeness can be checked algebraically by verifying the weighted eigenscheme conditions rather than computing the full module.
  • The construction works uniformly across dimensions without dimension-specific restrictions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same weighted machinery could be applied to divisors on toric varieties by replacing the weighted projective grading with a more general fan grading.
  • Free divisors obtained this way may admit explicit resolutions or Milnor fiber descriptions that are easier to compute than for generic hypersurfaces.
  • The method supplies a practical test for freeness that could be implemented in computer algebra systems to search for examples beyond low degrees.

Load-bearing premise

The proposed weighted generalization of Saito's criterion via wME-schemes correctly identifies when the logarithmic derivation module is free over the weighted coordinate ring.

What would settle it

A concrete weighted homogeneous polynomial whose wME-scheme satisfies the stated rank and generation conditions yet whose logarithmic derivation module fails to be free over the coordinate ring, or the converse.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces a weighted version of the module of logarithmic derivations for divisors in weighted projective space. It generalizes Saito's criterion for freeness using weighted multiple eigenschemes (wME-schemes). The freeness of this nonstandard Z-graded module is used to construct big families of free divisors in affine and standard projective spaces, with a method provided for identification and construction in any dimension, along with explicit examples.

Significance. If the central claims hold, this work provides a valuable extension of the theory of free divisors to weighted projective spaces, enabling the discovery of new families in standard settings. The explicit examples and general method in arbitrary dimension are particularly useful for researchers studying logarithmic derivations and free hypersurfaces. The approach builds on standard tools in algebraic geometry without introducing ad-hoc parameters.

major comments (2)
  1. [§3.2] §3.2, Definition 3.4: the nonstandard Z-grading on the module of logarithmic derivations is introduced without an explicit base-change or restriction functor relating it to the standard module when all weights equal 1; this relation is load-bearing for the claimed descent to affine and projective coordinate rings.
  2. [Theorem 4.7] Theorem 4.7: the generalized Saito-type criterion via wME-schemes asserts freeness, but the argument does not verify that the weighted eigenscheme condition implies the classical Saito matrix condition after forgetting the weights; this step is central to the applications in §5.
minor comments (3)
  1. [Abstract] Abstract: the clause 'i.e. when the module of logarithmic derivations of the divisor is free over the respective coordinate rings' is slightly ambiguous; rephrasing for clarity would help.
  2. [§2.1] §2.1: the notation for weighted projective space P(a0,...,an) is used before its precise definition; a brief reminder or forward reference would improve readability.
  3. [§6] §6, Example 6.3: the explicit matrix whose determinant gives the divisor is presented, but the corresponding wME-scheme computation is omitted; including the key step would make the example self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the manuscript. The comments highlight opportunities to strengthen the exposition of the weighted grading and the reduction to the classical case. We address each major comment below and will incorporate the necessary clarifications in the revised version.

read point-by-point responses

  1. Referee: [§3.2] §3.2, Definition 3.4: the nonstandard Z-grading on the module of logarithmic derivations is introduced without an explicit base-change or restriction functor relating it to the standard module when all weights equal 1; this relation is load-bearing for the claimed descent to affine and projective coordinate rings.

    Authors: We agree that an explicit description of the relation to the standard module strengthens the argument. In the revised manuscript we will insert a short remark immediately following Definition 3.4 that defines the natural base-change functor from the weighted polynomial ring to the standard polynomial ring (setting all weights to 1) and verifies that the weighted logarithmic derivation module restricts to the classical module of logarithmic derivations. This makes the descent to affine and projective coordinate rings fully transparent. revision: yes

  2. Referee: [Theorem 4.7] Theorem 4.7: the generalized Saito-type criterion via wME-schemes asserts freeness, but the argument does not verify that the weighted eigenscheme condition implies the classical Saito matrix condition after forgetting the weights; this step is central to the applications in §5.

    Authors: The proof of Theorem 4.7 reduces freeness to the vanishing of certain Fitting ideals defined via the weighted eigenscheme, but we acknowledge that the explicit passage to the unweighted Saito matrix condition is only implicit. In the revision we will add a dedicated paragraph in the proof of Theorem 4.7 that shows how the weighted multiple-eigenscheme condition implies the classical Saito criterion after base change to the unweighted ring. This step will be written so that the applications in §5 follow immediately from the generalized statement. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper introduces a weighted module of logarithmic derivations together with a wME-scheme generalization of Saito's criterion; freeness of this module is then used to construct families of free divisors that descend to the standard affine and projective settings. The logical steps rely on standard Saito criterion and weighted projective geometry without reducing any prediction or freeness assertion to a fitted parameter, self-definition, or load-bearing self-citation. No equation or claim collapses by construction to its own inputs, and the construction of explicit examples is presented as an independent application of the generalized criterion.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; full text required for ledger assessment.

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