  We compute Hochschild cohomology of projective hypersurfaces starting from
the Gerstenhaber-Schack complex of the (restricted) structure sheaf. We are
particularly interested in the second cohomology group and its relation with
deformations. We show that a projective hypersurface is smooth if and only if
the classical HKR decomposition holds for this group. In general, the first
Hodge component describing scheme deformations has an interesting inner
structure corresponding to the various ways in which first order deformations
can be realized: deforming local multiplications, deforming restriction maps,
or deforming both. We make our computations precise in the case of quartic
hypersurfaces, and compute explicit dimensions in many examples.
