What are some surprising or memorable examples in algebraic geometry, suitable for a course I'll be teaching on chapters 1-2 of Hartshorne (varieties, introductory schemes)?
I'd prefer examples that are unusual or nonstandard, as I already know many of the standard ones (27 lines on a cubic surface, etc). To illustrate the sorts of things I am looking for, here are some examples that would have been useful answers if I had not already thought of them:
The proof of the Kakeya conjecture for finite fields.
Free sheaves need not be projective
The Hilbert scheme of m points on an n dimensional variety can have dimension larger than mn.
The scheme of nilpotent matrices is not reduced.
The 1-line proof of Pascal's theorem from Bezout's theorem.
Resolution of the Whitney umbrella
The related threads http://mathoverflow.net/questions/28496/what-should-be-learned-in-a-first-serious-schemes-course and http://mathoverflow.net/questions/20112/interesting-results-in-algebraic-geometry-accessible-to-3rd-year-undergraduates also have some good examples, such as Grassmannians.
Added later: Thanks for all the examples; I'm embarrassed to admit that some of them are counterexamples to statements I would have guessed were true. Over the next few weeks I will gradually add answers below (with credit to those who suggested them) to my draft course notes (These notes still need a lot of corrections/expansion/rewriting; they should have reached a more stable state by Dec 2010)