Invariants of CR-manifolds associated with the tangent quadric

We construct a system of $\operatorname{CR}$-invariants of a manifold generated by projective invariants of the tangent quadric. We present a description of the group of projective diffeomorphisms of a quartic. We also estimate the degree of a rational mapping of a quartic. The description problem for subgroups of a Cremona group of bounded degree is posed.