Embedding of a maximal curve in a Hermitian variety

1999/11/08 by Gabor Korchmaros, Fernando Torres
Mathematics · #math.AG #msc:11G20 #msc:11G #msc:11 #msc:14G15 #msc:14G #msc:14

paper · pdf

published as Compositio Math. 128 (2001), 95--113 · 18 pages, LaTex2e

arxiv created 1999/11/08 · arxiv updated 2009/11/30

Abstract

Let X be a projective geometrically irreducible non-singular algebraic curve defined over a finite field F of order q². 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 P₀∈ X(F), let πbe the morphism arising from the linear series D:=|(q+1)P₀|, and let N:=dim(D). It is known that N≥ 2 and that πis independent of P₀ whenever X is F-maximal. The following theorems will be proved: Theorem 0.1: If X is F-maximal, then π:X→ 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( F); Theorem 0.2: If X is F-maximal, then it is F-isomorphic to a curve Y in P^M( F), with 2≤ M≤ N, such that Y has degree q+1 and lies on a non-degenerate Hermitian variety defined over F of ¶^M( F). Furthermore, Aut_F(X) is isomorphic to a subgroup of the projective unitary group PGU(M+1,q²); Theorem 0.3: If X is F-birational to a curve Y embedded in P^M( F) such that Y has degree q+1 and lies on a non-degenerate Hermitian variety defined over F of P^M( F), then X is F-maximal and X is F-isomorphic to Y.

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