Covered by lines and Conic connected varieties

Abstract

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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 basic results by Ein and Zak.