Symmetric differentials on hypersurfaces in weighted projective three space

We investigate symmetric differentials on hypersurfaces in weighted projective three space. For weighted degree $d>\sum a_i$, we prove twisted vanishing for smooth hypersurfaces and the corresponding reflexive vanishing for well-formed quasi-smooth hypersurfaces. In the opposite direction, using the quotient-singularity criterion of Asega–De Oliveira–Weiss together with weighted Kummer and Segre constructions, we produce singular weighted hypersurfaces whose minimal resolutions have big cotangent bundle. These include examples in $\text{PP}(1,1,1,r)$ for every $r\ge 2$. Combining the vanishing and bigness results with simultaneous resolution yields deformation-equivalent smooth projective surfaces for which the symmetric plurigenera jump from zero in every positive order to cubic asymptotic growth.