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 $10$.

Comment: 28 pages, 7 figures

• 11G35 - Varieties over global fields
• 14F10 - Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
• 14J17 - Singularities of surfaces or higher-dimensional varieties
• 14M25 - Toric varieties, Newton polyhedra, Okounkov bodies
• 52B20 - Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)

• 1 zbMATH Open