Afsaneh Mehran

"Bend-and-Break" Lemmas and Frobenius trick

The original Bend-and-Break lemma says that if a curve deforms non-trivially on a projective variety, while keeping a point fixed, then at some point it must break up with at least one rational component containing the fixed point. However, to ensure that the curve deforms fixing the point, we need to check some numerical conditions. In the case of a Fano variety we can fulfill those conditions, using characteristic p methods, and therefore prove that in all characteristics, it is covered by rational curves.