On the genus of a maximal curve

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dc.creatorKorchmaros, Gabor
dc.creatorTorres, Fernando
dc.date2000-08-26
dc.date.accessioned2026-07-07T04:37:00Z
dc.date.available2026-07-07T04:37:00Z
dc.descriptionPrevious 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.
dc.description23 pages, LaTex2e, ps file available at http://www.ime.unicamp.br/~ftorres
dc.identifierhttps://arxiv.org/abs/math/0008202
dc.identifierhttp://arxiv.org/abs/math/0008202
dc.identifierMath. Ann. 323(3) (2002), 589--608
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/59804
dc.subjectAlgebraic Geometry
dc.subject14G, 11G
dc.titleOn the genus of a maximal curve
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