When does a curve move on a surface, especially over finite fields? -- Burt Totaro

We start with some general discussion of the cone of curves on a projective variety. There are classical examples of bad behavior of the cone of curves, by Zariski and others (curves that look like they should move, but don't). Sean Keel showed that curves behave better on varieties over finite fields. Nonetheless, we give a negative answer to Keel's main open question about curves on varieties over finite fields.