[Paper Review] Embedding of a maximal curve in a Hermitian variety

Gábor Korchmáros, Fernando Torres|ArXiv.org|Nov 8, 1999

Algebraic Geometry and Number Theory9 references4 citations

TL;DR

This paper establishes that every F-maximal curve over a finite field of order $q^2$ is F-isomorphic to a non-singular curve of degree $q+1$ embedded in a non-degenerate Hermitian variety in projective space $\mathbb{P}^M$ for some $M \leq N$, where $N = \dim |(q+1)P_0|$. The key result shows that such curves are geometrically equivalent to curves lying on Hermitian varieties, and their automorphism groups embed into the projective unitary group $\mathrm{PGU}(M+1,q^2)$, providing a complete geometric characterization of maximal curves via Hermitian embeddings.

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 πbe 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 πis independent of P_0 whenever X is F-maximal. The following theorems will be proved: Theorem 0.1: If X is F-maximal, then π:X o π(X) is a F-isomorphism. The non-singular model π(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 ¶^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.

Motivation & Objective

• To characterize F-maximal curves over finite fields of order $q^2$ through their geometric embeddings in Hermitian varieties.
• To determine the conditions under which a curve is F-isomorphic to a curve lying on a non-degenerate Hermitian variety.
• To establish a correspondence between the automorphism group of a maximal curve and a subgroup of the projective unitary group $\mathrm{PGU}(M+1,q^2)$.

Proposed method

• Use the linear series $D = |(q+1)P_0|$ for a rational point $P_0$ on the curve $X$, defining a morphism $\pi: X \to \pi(X)$.
• Analyze the degree and dimension $N = \dim D$ of the image $\pi(X)$, showing it has degree $q+1$.
• Prove that $\pi(X)$ lies on a Hermitian variety defined over $\mathbb{F}_{q^2}$ in $\mathbb{P}^N(\bar{\mathbb{F}}_{q^2})$.
• Establish that the morphism $\pi$ is an F-isomorphism when $X$ is F-maximal.
• Construct an embedding of $X$ into $\mathbb{P}^M(\bar{\mathbb{F}}_{q^2})$ for $2 \leq M \leq N$ such that the image lies on a non-degenerate Hermitian variety.
• Use the structure of the automorphism group $\mathrm{Aut}_F(X)$ to show it embeds into $\mathrm{PGU}(M+1,q^2)$.

Experimental results

Research questions

• RQ1 Under what conditions is an F-maximal curve F-isomorphic to a curve lying on a Hermitian variety in projective space?
• RQ2 Can every F-maximal curve be embedded as a non-singular curve of degree $q+1$ in a non-degenerate Hermitian variety defined over $\mathbb{F}_{q^2}$?
• RQ3 How does the automorphism group of an F-maximal curve relate to the projective unitary group $\mathrm{PGU}(M+1,q^2)$?
• RQ4 Is the morphism $\pi: X \to \pi(X)$ arising from $|(q+1)P_0|$ an F-isomorphism when $X$ is F-maximal?
• RQ5 What is the minimal dimension $M$ such that an F-maximal curve can be embedded in $\mathbb{P}^M(\bar{\mathbb{F}}_{q^2})$ on a non-degenerate Hermitian variety?

Key findings

• The morphism $\pi: X \to \pi(X)$ is an F-isomorphism when $X$ is F-maximal, and $\pi(X)$ has degree $q+1$.
• The image $\pi(X)$ lies on a Hermitian variety defined over $\mathbb{F}_{q^2}$ in $\mathbb{P}^N(\bar{\mathbb{F}}_{q^2})$, where $N = \dim |(q+1)P_0|$.
• Every F-maximal curve $X$ is F-isomorphic to a curve $Y$ of degree $q+1$ lying on a non-degenerate Hermitian variety in $\mathbb{P}^M(\bar{\mathbb{F}}_{q^2})$ for some $M$ with $2 \leq M \leq N$.
• The automorphism group $\mathrm{Aut}_F(X)$ is isomorphic to a subgroup of the projective unitary group $\mathrm{PGU}(M+1,q^2)$.
• If a curve $X$ is F-birational to a curve $Y$ of degree $q+1$ lying on a non-degenerate Hermitian variety over $\mathbb{F}_{q^2}$, then $X$ is F-maximal and F-isomorphic to $Y$, establishing a converse to the embedding result.