Remarks on plane maximal curves

Some new results on plane $\fq$-maximal curves are stated and proved. By \cite{r-sti}, the degree $d$ of a plane $\fq$-maximal curve is less than or equal to $q+1$ and equality holds if and only if the curve is $\fq$-isomorphic to the Hermitian curve. We show that $d\leq q+1$ can be improved to $d\leq (q+2)/2$ apart from the case $d=q+1$ or $q\leq 5$. This upper bound turns out to be sharp for $q$ odd. In \cite{carbonne-henocq} it was pointed out that some Hurwitz curves are plane $\fq$-maximal curves. Here we prove that (\ref{eq1.2}) is the necessary and sufficient condition for a Hurwitz curve to be $\fq$-maximal. We also show that this criterium holds true for the $\fq$-maximality of a wider family of curves.