Covered by Lines and Conic Connected Varieties

Abstract

We study some properties of an embedded variety covered by lines and give a numerical criterion ensuring the existence of a singular conic through two of its general points. We show that our criterion is sharp. Conic-connected, covered by lines, QEL, LQEL, prime Fano, defective, and dual defective varieties are closely related. We study some relations between the above mentioned classes of objects using celebrated results by Ein and Zak.