Remarks on plane maximal curves

Details Remarks on plane maximal curves Remarks on plane maximal curves

2000-03-28

arxiv.org/abs/math/0003179v1

Acta Arith. 98(2) (2001), 165--179

Mathematics

Algebraic Geometry

14 pages, LaTex2e

Scientific paper

Some new results on plane F_{q^2}-maximal curves are stated and proved. It is known that the degree d of such curves is upper bounded by q+1 and that d=q+1 if and only if the curve is F_{q^2}-isomorphic to the Hermitian. We show that d\le q+1 can be improved to d\le (q+2)/2 apart from the case d=q+1 or q\le 5. This upper bound turns out to be sharp for q odd. We also study the maximality of Hurwitz curves of degree n+1. We show that they are F_{q^2}-maximal if and only if (q+1) divides (n^2-n+1). Such a criterion is extended to a wider family of curves.

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