The log canonical threshold and rational singularities
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| author | Cluckers, Raf Kollár, János Mustaţă, Mircea |
| author_facet | Cluckers, Raf Kollár, János Mustaţă, Mircea |
| contents | We show that if $f$ is a nonzero, noninvertible function on a smooth complex variety $X$ and $J_f$ is the Jacobian ideal of $f$, then ${\rm lct}(f,J_f^2)>1$ if and only if the hypersurface defined by $f$ has rational singularities. Moreover, if it does not have rational singularities, then ${\rm lct}(f,J_f^2)={\rm lct}(f)$. We give two proofs, one relying on arc spaces and one that goes through the inequality $\widetildeα(f)\geq{\rm lct}(f,J_f^2)$, where $\widetildeα(f)$ is the minimal exponent of $f$. In the case of a polynomial over $\overline{\mathbf{Q}}$, we also prove an analogue of this latter inequality, with $\widetildeα(f)$ replaced by the motivic oscillation index ${\rm moi}(f)$. We also show a part of Igusa's strong monodromy conjecture, for poles larger than $-{\rm lct}(f,J_f^2)$. We end with a discussion of lct-maximal ideals: these are ideals $I$ with the property that ${\rm lct}(I)<{\rm lct}(J)$ for every $J$ with $I\subsetneq J$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_08425 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The log canonical threshold and rational singularities Cluckers, Raf Kollár, János Mustaţă, Mircea Algebraic Geometry 14B05, 14E18, 14J17, 11L07 We show that if $f$ is a nonzero, noninvertible function on a smooth complex variety $X$ and $J_f$ is the Jacobian ideal of $f$, then ${\rm lct}(f,J_f^2)>1$ if and only if the hypersurface defined by $f$ has rational singularities. Moreover, if it does not have rational singularities, then ${\rm lct}(f,J_f^2)={\rm lct}(f)$. We give two proofs, one relying on arc spaces and one that goes through the inequality $\widetildeα(f)\geq{\rm lct}(f,J_f^2)$, where $\widetildeα(f)$ is the minimal exponent of $f$. In the case of a polynomial over $\overline{\mathbf{Q}}$, we also prove an analogue of this latter inequality, with $\widetildeα(f)$ replaced by the motivic oscillation index ${\rm moi}(f)$. We also show a part of Igusa's strong monodromy conjecture, for poles larger than $-{\rm lct}(f,J_f^2)$. We end with a discussion of lct-maximal ideals: these are ideals $I$ with the property that ${\rm lct}(I)<{\rm lct}(J)$ for every $J$ with $I\subsetneq J$. |
| title | The log canonical threshold and rational singularities |
| topic | Algebraic Geometry 14B05, 14E18, 14J17, 11L07 |
| url | https://arxiv.org/abs/2202.08425 |