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arXiv · 2509.16933

Quasihomogeneous isolated singularities in terms of syzygies and foliations

Hamid Hassanzadeh·Abbas Nasrollah Nejad·Aron Simis
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Abstract

One considers quasihomogeneous isolated singularities of hypersurfaces in arbitrary dimensions through the lenses of three apparently quite apart themes: syzygies, singularity invariants, and foliations. In the first of these, one adds to the well-known result of Saito's a syzygy-theoretic characterization of a quasihomogeneous singularity affording an effective computational criterion. In the second theme, one explores the Milnor-Tjurina difference number from a commutative algebra viewpoint. Building on the Briancon-Skoda theorem and exponent, we extend previously known inequalities by Dimca and Greuel to arbitrary dimension and provide algebraic formulas involving the syzygy-theoretic part and reduction exponents. In the last theme one recovers and bring up to an algebraic light a result of Camacho and Movasati by establishing a couple of characterizations of quasihomogeneous isolated singularities in terms of the generators of the module of invariant vector fields.

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Hamid Hassanzadeh, Abbas Nasrollah Nejad, Aron Simis. 2025-09-21. Quasihomogeneous isolated singularities in terms of syzygies and foliations. https://arxiv.org/abs/2509.16933

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