  We consider finite-dimensional Hopf algebras $u$ which admit a smooth
deformation $U\to u$ by a Noetherian Hopf algebra $U$ of finite global
dimension. Examples of such Hopf algebras include small quantum groups over the
complex numbers, restricted enveloping algebras in finite characteristic, and
Drinfeld doubles of height $1$ group schemes. We provide a means of analyzing
(cohomological) support for representations over such $u$, via the singularity
categories of the hypersurfaces $U/(f)$ associated to functions $f$ on the
corresponding parametrization space. We use this hypersurface approach to
establish the tensor product property for cohomological support, for the
following examples: functions on a finite group scheme, Drinfeld doubles of
certain height 1 solvable finite group schemes, bosonized quantum complete
intersections, and the small quantum Borel in type $A$.
