I decided to spend this summer working through exercises in Hartshorne, and I found myself frustrated by the way I was solving one of them, specifically IV.2.3 on pp. 304-305. No, I'm not asking for a solution to the problem --- I almost have the whole thing solved, as far as I can tell --- what worries me is that I think I'm thinking about this stuff the wrong way.
The problem is about the map from a projective plane curve in characteristic 0 to its dual curve, and it has 8 parts, which ask you to prove things like the fact that bitangents on the original curve correspond to nodes on the dual. I managed to solve all of them except the one about ordinary inflection points on X giving ordinary cusps on X*, but all my solutions involved picking affine charts, finding an ugly formula for the map in question, and computing first and second partial derivatives, and it's all very long and messy and doesn't give any clue as to what's going on until you get to the end and see that the thing you have is 0 or whatever if and only if the condition in the problem is met for some magical reason.
I feel like there must be a more "high-brow" way to approach this object, and the fact that I haven't been able to come up with one seems to speak to the sort of backward way I've been learning about the subject (I just took a class that was very good and covered a lot but was almost completely devoid of examples). There must be some approach to this that actually uses all the machinery that's developed in the rest of the book. Is there a satisfying, pretty way to deal with this thing that I'm missing, something that would tell me why these relationships hold and not just that they hold?