  Let $X$ be a smooth Fano threefold over an algebraically closed field of
positive characteristic. Assume that $|-K_X|$ is very ample and each of the
index and the Picard number is equal to one. We prove that $3 \leq g \leq 12$
and $g \neq 11$ for the genus $g$ of $X$. Moreover, we show that there exists
no smooth curve on $X$ along which the blowup is Fano.
