Resolution of singularities.
That isn't really an answer to the question - I don't think it's necessary in a first course, but I do think resolution should be rotated in on a regular basis, which requires the annual core to be small enough to make room for it. I think it's valuable not just to teach the material somewhere in the curriculum, but to put it in an introductory course, to emphasize that it is elementary and not impossibly difficult. Also, to contrast with the Grothendieck-flavored majority.
