On manifolds swept out by high dimensional quadrics.

I outline a completely different, much shorter, proof of a substantial generalization of the main result of Y. Kachi and E. Sato (Segre's Reflexivity and an Inductive Characterization of
Hyperquadrics, Mem. Amer. Math. Soc., vol. 160, no. 763, 2002).
It states that embedded projective n-folds swept out by quadrics of
dimension at least [n/2]+2 ([ ] denoting the integral part) are either scrolls or hyperquadric fibrations, which are also Mori
contractions (joint work with P. Ionescu).