Connectedness Bertini Theorem via numerical equivalence

Let X be an irreducible projective variety and let f : X → ℙn be a morphism.We give a new proof of
the fact that the preimage of any linear variety of dimension k ≥ n+1−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.