Multiplier ideals of normal surface singularities
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910655561334784 |
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| author | Koltai, László László, Tamás Némethi, András |
| author_facet | Koltai, László László, Tamás Némethi, András |
| contents | We study the multiplier ideals and the corresponding jumping numbers and multiplicities $\{m(c)\}_{c\in \mathbb{R}}$ in the following context: $(X,o)$ is a complex analytic normal surface singularity, ${\mathfrak a}\subset \mathcal{O}_{X,o}$ is an ${\mathfrak m}_{X,o}$--primary ideal, $ϕ:\widetilde{X}\to X$ is a log resolution of $\mathfrak{a}$ such that $\mathfrak{a}\mathcal{O}_{\widetilde{X}}=\mathcal{O}_{\widetilde{X}}(-F)$, for some nonzero effective divisor $F$ supported on $ϕ^{-1}(0)$. We show that $\{m(c)\}_{c>0}$ is combinatorially computable from $F$ and the resolution graph $Γ$ of $ϕ$, and we provide several formulae. We also extend Budur's result (valid for $(X,o)=(\mathbb{C}^2,0)$), which makes an identification of $\sum_{c\in[0,1]}m(c)t^c$ with a certain Hodge spectrum. In our general case we use Hodge spectrum with coefficients in a mixed Hodge module. We show that $\{m(c)\}_{c\leq 0}$ usually depends on the analytic type of $(X,o)$. However, for some distinguished analytic types we determine it concretely. E.g., when $(X,o)$ is weighted homogeneous (and $F$ is associated with the central vertex), we recover $\sum_cm(c)t^c$ from the Poincaré series of $(X,o)$ and when $(X,o)$ is a splice quotient then we recover $\sum_cm(c)t^c$ from the multivariable topological Poincaré (zeta) function of $Γ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13413 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multiplier ideals of normal surface singularities Koltai, László László, Tamás Némethi, András Algebraic Geometry 14B05, 14Fxx, 32S05, 32S10, 32S25 We study the multiplier ideals and the corresponding jumping numbers and multiplicities $\{m(c)\}_{c\in \mathbb{R}}$ in the following context: $(X,o)$ is a complex analytic normal surface singularity, ${\mathfrak a}\subset \mathcal{O}_{X,o}$ is an ${\mathfrak m}_{X,o}$--primary ideal, $ϕ:\widetilde{X}\to X$ is a log resolution of $\mathfrak{a}$ such that $\mathfrak{a}\mathcal{O}_{\widetilde{X}}=\mathcal{O}_{\widetilde{X}}(-F)$, for some nonzero effective divisor $F$ supported on $ϕ^{-1}(0)$. We show that $\{m(c)\}_{c>0}$ is combinatorially computable from $F$ and the resolution graph $Γ$ of $ϕ$, and we provide several formulae. We also extend Budur's result (valid for $(X,o)=(\mathbb{C}^2,0)$), which makes an identification of $\sum_{c\in[0,1]}m(c)t^c$ with a certain Hodge spectrum. In our general case we use Hodge spectrum with coefficients in a mixed Hodge module. We show that $\{m(c)\}_{c\leq 0}$ usually depends on the analytic type of $(X,o)$. However, for some distinguished analytic types we determine it concretely. E.g., when $(X,o)$ is weighted homogeneous (and $F$ is associated with the central vertex), we recover $\sum_cm(c)t^c$ from the Poincaré series of $(X,o)$ and when $(X,o)$ is a splice quotient then we recover $\sum_cm(c)t^c$ from the multivariable topological Poincaré (zeta) function of $Γ$. |
| title | Multiplier ideals of normal surface singularities |
| topic | Algebraic Geometry 14B05, 14Fxx, 32S05, 32S10, 32S25 |
| url | https://arxiv.org/abs/2407.13413 |