Purity Results on $F$-crystals
  For an $F^n$-crystal $\mathfrak C$ over a reduced locally Noetherian $\mathbb F_p$-scheme $S$, Vasiu first obtained the purity of the Newton polygon strata of $S$ defined by $\mathfrak C$. Deligne later obtained the purity of the $p$-rank strata of $S$ defined by $\mathfrak C$. We prove a stronger result that for every fixed point $P_0$ in the $xy$-coordinate plane, the reduced locally closed subscheme $S_{P_0}=\{s\in S| NP(\mathfrak C_s)\mbox{ has }P_0\mbox{ as a break point}\}$ of $S$ is pure in $S$ (A weaker result was known for the weak purity of $S_{p_0}$). 
Jinghao Li
math.AG