Previous results on genera g of F_{ q^2}-maximal curves are improved: (1) Either g\leq (q^2-q+4)/6, or g=\lfloor(q-1)^2/4\rfloor, or g=q(q-1)/2; (2) The hypothesis on the existence of a particular Weierstrass point in \cite{ at} is proved; (3) For q\equiv 1\pmod{ 3}, q\ge 13, no F_{ q^2}-maximal curve of genus (q-1)(q-2)/3 exists; (4) For q\equiv 2\pmod{ 3}, q\ge 11, the non-singular F_{ q^2}-model of the plane curve of equation y^q+y=x^{ (q+1)/3} is the unique F_{ q^2}-maximal curve of genus g=(q-1)(q-2)/6; (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 F_{ q^2}-maximal curve of genus g=(q-1)(q-3)/8. For q\equiv 3\pmod{ 4}, q\ge 19, there are exactly two F_{ q^2}-maximal curves of genus g=(q-1)(q-3)/8, namely the above Fermat curve and the non-singular F_{ q^2}-model of the plane curve of equation y^q+y=x^{ (q+1)/4}. 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.
Categories: math.AG