Embedding of a maximal curve in a Hermitian variety

Abstract

Let X be a projective geometrically irreducible non-singular algebraic curve defined over a finite field F of order $q^2$. If the number of F-rational points of X satisfies the Hasse-Weil upper bound, then X is said to be F-maximal. For a point P_0\in X(F), let \pi be the morphism arising from the linear series D:=|(q+1)P_0|, and let N:=dim(D). It is known that N\ge 2 and that \pi is independent of P_0 whenever X is F-maximal. The following theorems will be proved: Theorem 0.1: If X is F-maximal, then \pi:X\to \pi(X) is a F-isomorphism. The non-singular model \pi(X) has degree q+1 and lies on a Hermitian variety defined over F of P^N(\bar F); Theorem 0.2: If X is F-maximal, then it is F-isomorphic to a curve Y in P^M(\bar F), with 2\le M\le N, such that Y has degree q+1 and lies on a non-degenerate Hermitian variety defined over F of \P^M(\bar F). Furthermore, Aut_F(X) is isomorphic to a subgroup of the projective unitary group PGU(M+1,q^2); Theorem 0.3: If X is F-birational to a curve Y embedded in P^M(\bar F) such that Y has degree q+1 and lies on a non-degenerate Hermitian variety defined over F of P^M(\bar F), then X is F-maximal and X is F-isomorphic to Y.