On the critical loci of finite maps

Consider a rational function f(z) of degree n. To f(z), we can associate its set of critical points, namely the points z where the derivative vanishes. A quick count shows that the number of critical points is the same as the number of coefficients needed to describe f(z), and therefore, given a set of points, we should expect a finite number of rational functions with that set as its critical locus. It has long been known that this is indeed the case, and that this number is the n-th Catalan number.

I will explore the analogue of this phenomenon for algebraic varieties of arbitrary dimension. I will categorize the cases that lead to a finite number as above, and say something about what these numbers might be. Much of this work is in progress and conjectural, and all of it is joint with Anand Patel.