Nils Bruin ; Nathan Ilten ; Zhe Xu - Local Euler characteristics of $A_n$-singularities and their application to hyperbolicity

epiga:12665 - Épijournal de Géométrie Algébrique, February 6, 2025, Volume 9 - https://doi.org/10.46298/epiga.2025.12665
Local Euler characteristics of $A_n$-singularities and their application to hyperbolicityArticle

Authors: Nils Bruin ; Nathan Ilten ; Zhe Xu

    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 $10$.


    Volume: Volume 9
    Published on: February 6, 2025
    Accepted on: May 24, 2024
    Submitted on: December 7, 2023
    Keywords: Mathematics - Algebraic Geometry,Mathematics - Number Theory,Primary 14J17, 14M25, Secondary 14F10, 52B20, 11G35

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