  Let $k$ be an algebraically closed field of characteristic $p >0$. Suppose $g
\geq 3$ and $0 \leq f \leq g$. We prove there is a smooth projective $k$-curve
of genus $g$ and $p$-rank $f$ with no non-trivial automorphisms. In addition,
we prove there is a smooth projective hyperelliptic $k$-curve of genus $g$ and
$p$-rank $f$ whose only non-trivial automorphism is the hyperelliptic
involution. The proof involves computations about the dimension of the moduli
space of (hyperelliptic) $k$-curves of genus $g$ and $p$-rank $f$ with extra
automorphisms.
