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Let k be an algebraically closed field of characteristic p > 0. Suppose g ≥ 3 and 0 ≤ f ≤ 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 extradoi:10.1307/mmj/1231770361 fatcat:pztouwzdsjcaxe22hgytok44za