This paper, joint with A. Deopurkar and E. Duryev, proves a result in classical projective geometry. I was initially motivated, (and still am) by the following purposefully vague question: In projective geometry, we have some standard geometric constructions. For instance, we often take intersections with a hyperplane. Or we project a variety to another projective space. In the process, we get several auxiliary geometric figures on the original variety -- do these figures "vary maximally" with the parameters underlying the construction? For instance, a general projection of a variety to a projective space of the same dimension produces a ramification divisor. Does the divisor vary maximally with the source of projection?

We thoroughly investigate the effect on ramification divisors of perturbations of projections, and in the process we discover some rather peculiar counter-examples to the maximal variation expectation. Along the way, we also discover a new classification result characterizing rational normal scrolls, and we find that each rational normal scroll is host to a very challenging (and still wide-open) Projection-Ramification enumerative problem. Computer experiments tell us that these problems promise beautiful answers, generalizing the well-known cameo appearance of Catalan numbers in the extrinsic geometry of rational normal curves.