It's a classical theorem in complex algebraic geometry that a smooth surface which is a "generic" complete intersection in projective space has all its divisors coming from intersections with hypersurfaces in the ambient space. I will discuss the case of hypersurfaces in P^3 and explain how this result can be given a cohomological interpretation. I will then sketch the key ideas in the proof and how a local monodromy computation plays a central role. The talk will include an overview of the results we need from the theory of l-adic etale cohomology. This talk combines ideas scattered throughout SGA 7 and Deligne's Weil II paper.