Eugene Eisenstein

Fano -------> Rationally Connected


This is the continuation of last week's lecture on bend and break. Mori's bend and break technique are a collection of lemmas that all revolve around a simple and very geometric idea: a curve that bends with some appropriate number of points fixed must, in the limit, break off into a rational component and some other curves. Having explained some of the reasons the lemmas are important last week, we will apply them to show that Fano varieties have a rational curve through every point. Time permitting, I will outline the proof that Fano varieties are rationally connected (there is a rational curve through any two points).