This is a talk about a problem in topology that comes out of algebraic geometry. We consider a smooth algebraic surface (a special type of 4-manifold) equipped with a family of embedded Riemann surfaces- a basic example is the family of “smooth plane curves” inside the complex projective plane. This leads to some problems in topology: first, to describe the topology of the “discriminant complement” that parametrizes the family, and secondly to describe its “monodromy representation”, a subgroup of the mapping class group of the Riemann surfaces in the family. I will introduce this problem, present some examples, and state some general results. Portions are joint with Ishan Banerjee.