Algebraic hyperbolicity of very general hypersurfaces in weighted projective spaces
Jiahe Wang
TL;DR
The paper addresses when very general hypersurfaces in weighted projective spaces with isolated singularities are algebraically hyperbolic. It extends the normal-bundle technique to this singular setting, using section-dominating collections to bound the degree of the normal bundle and derive a genus bound relative to a chosen ample divisor. A key outcome is a universal Theta-bound: if $m > \Theta$, where $\Theta$ is defined by a maximum over toric strata, then the hypersurface is algebraically hyperbolic outside the toric boundary; this yields explicit thresholds in dimension three and recovers known bounds in special weighted projective spaces. The results rely on toric resolutions, the Lazarsfeld–Mukai bundle framework, and the interplay between the toric boundary and the ambient Picard group, providing concrete criteria for hyperbolicity in singular ambient spaces with isolated singularities.
Abstract
We provide a bound for $m$ such that the zero locus of a very general section of an $m$-multiple of some ample line bundle on a weighted projective space with isolated singularities is algebraically hyperbolic.