Ideals generated by quadrics

Abstract

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 $d\ge 3$, when $K=\mathbb{R}$, we give structure theorems for all ideals of codimension $d$ minimally generated by $\left(\genfrac{}{}{0ex}{}{d+1}{2}\right)-1$ quadrics. For arbitrary fields $K$, we prove a polarized version.