A vector bundle on $\mathbb{P}^n$ is said to be $r$-regular if $$H^i(\mathbb{P}^n,F(r-i))=0$$ for all $i>0$. It is always true that if $F$ is $r$-regular and $G$ is $s$-regular (both vector bundles), that $F \otimes G$ is $r+s$ regular independent of characteristic.
In characteristic zero, this implies that $Sym^k F$ and $\Lambda^k F$ is $kr$ regular if $F$ is $r$ regular. This proof doesn't work if $p \le k$ where $p$ is the characteristic. But, is the conclusion known to be false? Are there any bounds on the regularity of symmetric powers and wedge powers of vector bundles in positive characteristic?
