Connectedness Bertini Theorem via numerical equivalence

Abstract

Let $X$ be an irreducible projective variety and let $f: X \rightarrow \mathbb{P}^n$ be a morphism. We give a new proof of the fact that the preimage of any linear variety of dimension $k \geq n+1-\operatorname{dim} f(X)$ is connected. We show that the statement is a consequence of the Generalized Hodge Index Theorem using easy numerical arguments that hold in any characteristic. We also prove the connectedness Theorem of Fulton and Hansen as an application of our main theorem.