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.

Spec Z[x]

Resolution of the Whitney umbrella

The related threads What should be learned in a first serious schemes course? and 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)

nota quirk of schemes. $\endgroup$10more comments