Tjurina spectrum and graded symmetry of missing spectral numbers
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908981317861376 |
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| author | Jung, Seung-Jo Kim, In-Kyun Saito, Morihiko Yoon, Youngho |
| author_facet | Jung, Seung-Jo Kim, In-Kyun Saito, Morihiko Yoon, Youngho |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17601 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tjurina spectrum and graded symmetry of missing spectral numbers Jung, Seung-Jo Kim, In-Kyun Saito, Morihiko Yoon, Youngho Algebraic Geometry 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. |
| title | Tjurina spectrum and graded symmetry of missing spectral numbers |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2411.17601 |