Embedding of a Maximal Curve in a Hermitian Variety

Let X be a projective geometrically irreducible non-singular algebraic curve defined over a finite field Fq2 of order q . If the number of Fq2 -rational points of X satisfies the Hasse-Weil upper bound, then X is said to be Fq2 -maximal. For a point P0 ∈ X (Fq2 ), let π be the morphism arising from the linear series D := |(q + 1)P0|, and let N := dim(D). It is known that N ≥ 2 and that π is independent of P0 whenever X is Fq2 -maximal. The following theorems will be proved: Theorem 0.1. If X is Fq2-maximal, then π : X → π(X ) is a Fq2-isomorphism. The non-singular model π(X ) has degree q + 1 and lies on a Hermitian variety defined over Fq2 of P N (F̄q2 ). Theorem 0.2. If X is Fq2 -maximal, then it is Fq2-isomorphic to a curve Y in P M (F̄q2), with 2 ≤ M ≤ N , such that Y has degree q+1 and lies on a non-degenerate Hermitian variety defined over Fq2 of P M (F̄q2). Furthermore, AutF q2 (X) is isomorphic to a subgroup of the projective unitary group PGU(M + 1, q). Theorem 0.3. If X is Fq2-birational to a curve Y embedded in P (F̄q2) such that Y has degree q + 1 and lies on a non-degenerate Hermitian variety defined over Fq2 of P (F̄q2 ), then X is Fq2-maximal and X is Fq2-isomorphic to Y.