Tjurina spectrum and graded symmetry of missing spectral numbers
| Autor: | Jung, Seung-Jo, Kim, In-Kyun, Saito, Morihiko, Yoon, Youngho |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | For a hypersurface isolated singularity defined by $f$, the Steenbrink spectrum can be obtained as the Poincar\'e 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 ambient dimension) is bounded by $[(\mu-\tau)/2]$. We can moreover improve the estimate of Brian\c{c}on-Skoda exponent in the semisimple monodromy case. Comment: Some sections in arxiv::2406.06242 are moved to this paper, where they fit better; Remark after Cor. 2 improved |
| Databáze: | arXiv |
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