Local Euler characteristics of $A_n$-singularities and their application to hyperbolicity
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913162819796992 |
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| author | Bruin, Nils Ilten, Nathan Xu, Zhe |
| author_facet | Bruin, Nils Ilten, Nathan Xu, Zhe |
| contents | Wahl's local Euler characteristic measures the local contributions of a singularity to the usual Euler characteristic of a sheaf. Using tools from toric geometry, we study the local Euler characteristic of sheaves of symmetric differentials for isolated surface singularities of type $A_n$. We prove an explicit formula for the local Euler characteristic of the $m$th symmetric power of the cotangent bundle; this is a quasi-polynomial in $m$ of period $n+1$. We also express the components of the local Euler characteristic as a count of lattice points in a non-convex polyhedron, again showing it is a quasi-polynomial. We apply our computations to obtain new examples of algebraic quasi-hyperbolic surfaces in $\mathbb{P}^3$ of low degree. We show that an explicit family of surfaces with many singularities constructed by Labs has no genus $0$ curves for the members of degree at least $8$ and no curves of genus $0$ or $1$ for degree at least $10$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01722 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Local Euler characteristics of $A_n$-singularities and their application to hyperbolicity Bruin, Nils Ilten, Nathan Xu, Zhe Algebraic Geometry Number Theory Primary 14J17, 14M25, Secondary 14F10, 52B20, 11G35 Wahl's local Euler characteristic measures the local contributions of a singularity to the usual Euler characteristic of a sheaf. Using tools from toric geometry, we study the local Euler characteristic of sheaves of symmetric differentials for isolated surface singularities of type $A_n$. We prove an explicit formula for the local Euler characteristic of the $m$th symmetric power of the cotangent bundle; this is a quasi-polynomial in $m$ of period $n+1$. We also express the components of the local Euler characteristic as a count of lattice points in a non-convex polyhedron, again showing it is a quasi-polynomial. We apply our computations to obtain new examples of algebraic quasi-hyperbolic surfaces in $\mathbb{P}^3$ of low degree. We show that an explicit family of surfaces with many singularities constructed by Labs has no genus $0$ curves for the members of degree at least $8$ and no curves of genus $0$ or $1$ for degree at least $10$. |
| title | Local Euler characteristics of $A_n$-singularities and their application to hyperbolicity |
| topic | Algebraic Geometry Number Theory Primary 14J17, 14M25, Secondary 14F10, 52B20, 11G35 |
| url | https://arxiv.org/abs/2312.01722 |