On the genus of a maximal curve

Previous results on genera $g$ of $\fq$-maximal curves are improved:\begin{enumerate}\item[\rm(1)] $\text{Either} \ g\leq \lfloor (q^2-q+4)/6\rfloor\,, \ \text{or} \ g=\lfloor(q-1)^2/4\rfloor\,, \ \text{or} \ g=q(q-1)/2\,$.\item[\rm(2)] The hypothesis on the existence of a particular Weierstrass point in \cite{at} is proved.\item[\rm(3)] For $q\equiv 1\pmod{3}$, $q\ge 13$, no $\fq$-maximal curve of genus $(q-1)(q-2)/6$ exists.\item[\rm(4)] For $q\equiv 2\pmod{3}$, $q\ge 11$, the non-singular $\fq$-model of the plane curve of equation $y^q+y=x^{(q+1)/3}$ is the unique $\fq$-maximal curve of genus $g=(q-1)(q-2)/6$.\item[\rm(5)] Assume $\dim(\cD_\cX)=5$, and $\char(\fq)\geq 5$. For $q\equiv 1\pmod{4}$, $q\geq 17$, the Fermat curve of equation $x^{(q+1)/2}+y^{(q+1)/2}+1=0$ is the unique $\fq$-maximal curve of genus $g=(q-1)(q-3)/8$. For $q\equiv 3\pmod{4}$, $q\ge 19$, there are exactly two $\fq$-maximal curves of genus $g=(q-1)(q-3)/8$, namely the above Fermat curve and the non-singular $\fq$-model of the plane curve of equation $y^q+y=x^{(q+1)/4}$.\end{enumerate}The above results provide some new evidences on maximal curves in connection with Castelnuovo's bound and Halphen's theorem, especially with extremal curves; see for instance the conjecture stated in Introduction.