  The Jacobian ideal provides the set of infinitesimally trivial deformations
for a homogeneous polynomial, or for the corresponding complex projective
hypersurface. In this article, we investigate whether the associated linear
deformation is indeed trivial, and show that the answer is no in a general
situation. We also give a characterization of tangentially smoothable
hypersurfaces with isolated singularities. Our results have applications in the
local study of variations of projective hypersurfaces, complementing the global
versions given by J. Carlson and P. Griffiths, R. Donagi and the author, and in
the study of isotrivial linear systems on the projective space, showing that a
general divisor does not belong to an isotrivial linear system of positive
dimension.

Algebraic Geometry