Arcs and curves over a finite field

In [11], a new bound for the number of points on an algebraic curve over a finite field of odd order was obtained, and applied to improve previous bou nds on the size of a complete are not contained in a conic, Here, a similar approach is used to show that a complete are in a plane of even order q ha s size q+2 or q-root q+1 or less than q-2 root q+6. To obtain this result, first a new characterization of a Hermitian curve for any square q is given ; more precisely, it is shown that a curve of sufficiently low degree has a certain upper bound for the number of its rational points with equality oc curring in this bound only when the curve is Hermitian. Finally, another ap plication is given concerning the degree of the curve on which a unital can lie. (C) 1999 Academic Press.