Embedding of a maximal curve in a Hermitian variety (Q2745363)

Let \(X\) be a geometrically irreducible non-singular projective curve over a finite field \(\mathbb{F}_{q^2}\) of order \(q^2\). Such a curve \(X\) is said to be \(\mathbb{F}_{q^2}\)-maximal if \(\#(X(\mathbb{F}_{q^2}))\) attains the Hasse-Weil upper bound. For a point \(P_0\in X(\mathbb{F}_{q^2})\), let \(\pi:X\to\mathbb{P}^N\) be the morphism associated to the linear series \({\mathcal D}=|(q+1)P_0|\). One of the main results of this paper shows that the morphism \(\pi\) is a closed embedding. Moreover, the authors prove that the image lies on a Hermitian variety defined over \(\mathbb{F}_{q^2}\) of \(\mathbb{P}^N\). By projecting \(X\) to an appropriate linear subspace, they show that \(X\) admits a nonsingular model given by a curve defined over \(\mathbb{F}_{q^2}\) of degree \(q+1\) lying on a nondegenerate Hermitian variety over \(\mathbb{F}_{q^2}\) of \(\mathbb{P}^M\) with \(M\leq N\), and that this property characterizes \(\mathbb{F}_{q^2}\)-maximal curves.