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arXiv · 2411.17601

Tjurina spectrum and graded symmetry of missing spectral numbers

Seung-Jo Jung·In-Kyun Kim·Morihiko Saito·Youngho Yoon
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Abstract

For a hypersurface isolated singularity defined by a convergent power series $f$, the Steenbrink spectrum can be defined as the Poincaré polynomial of the graded quotients of the $V$-filtration on the Jacobian ring of $f$. The Tjurina subspectrum is defined by replacing the Jacobian ring with its quotient by the image of the multiplication by $f$. We prove that their difference (consisting of missing spectral numbers) has a canonical graded symmetry. This follows from the self-duality of the Jacobian ring, which is compatible with the action of $f$ as well as the $V$-filtration. It implies for instance that the number of missing spectral numbers which are smaller than $(n{+}1)/2$ (with $n$ the number of variables) is bounded by $[(μ{-}τ)/2]$. We can moreover improve the estimate of Briançon-Skoda exponent in the semisimple monodromy case.

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Seung-Jo Jung, In-Kyun Kim, Morihiko Saito, Youngho Yoon. 2026-04-21. Tjurina spectrum and graded symmetry of missing spectral numbers. https://arxiv.org/abs/2411.17601

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