Ideals generated by quadrics

Our purpose is to study the cohomological properties of the Rees algebras of a class of ideals generated by quadrics. For all such ideals $I\subset R = K[x,y,z]$ we give the precise value of depth $R[It]$ and decide whether the corresponding rational maps are birational. In the case of dimension $d \geq 3$, when $K=\mathbb{R}$, we give structure theorems for all ideals of codimension $d$ minimally generated by ${{d+1}\choose{2}}-1$ quadrics. For arbitrary fields $K$, we prove a polarized version.