Tjurina Number Jumps and Unimodal Hypersurface Singularities in Positive Characteristic

This paper generalizes existing methods to derive stronger bounds on the modality of hypersurface singularities. Our results demonstrate that each sudden jump in the extended Tjurina number necessarily increases the modality. Furthermore, we provide a full classification of unimodal isolated hypersurface singularities in characteristic p > 3 under contact equivalence.