Rigid curves on $\bar M_{0,n}$ and arithmetic breaks
  A result of Keel and McKernan states that a hypothetical counterexample to the F-conjecture must come from rigid curves on $\bar {M}_{0,n}$ that intersect the interior. We exhibit several ways of constructing rigid curves. In all our examples, a reduction mod p argument shows that the classes of the rigid curves that we construct can be decomposed as sums of F-curves. 
Ana-Maria Castravet and Jenia Tevelev
math.AG