Abstract
Let X be a projective, geometrically irreducible, non-singular, algebraic curve defined over a finite field F q 2 of order q 2. If the number of F q 2-rational points of X satisfies the Hasse–Weil upper bound, then X is said to be F q 2-maximal. For a point P 0 ∈ X(F q 2), let π be the morphism arising from the linear series D: = |(q + 1)P 0|, and let N: = dim(D). It is known that N ≥ 2 and that π is independent of P 0 whenever X is F q 2-maximal.
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References
Abdón, M. and Torres, F.: On maximal curves in characteristic two, Manuscripta Math. 99 (1999), 39-53.
Ballico E. and Cossidente, A: On the generic hyperplane section of curves in positive characteristic, J. Pure Appl. Algebra 102 (1995), 243-250.
Cossidente, A., Korchmáros, G. and Torres, F.: On curves covered by the Hermitian curve, J. Algebra 216 (1999), 56-76.
Cossidente, A., Korchmáros, G. and Torres, F.: Curves of large genus covered by the Hermitian curve, Comm. Algebra 28(10) (2000), 4707-4728.
Eisenbud, D. and Harris, J.: Curves in Projective Space, Les Presses de l'Université de Montréal, Montréal, 1982.
Fuhrmann, R., García, A. and Torres, F.: On maximal curves, J. Number Theory 67 (1997), 29-51.
Fuhrmann, R. and Torres, F.: On Weierstrass points and optimal curves, Rend. Circ. Mat. Palermo 51 (1998), 25-46.
Garcia, A. and Voloch, J. F.: Wronskians and independence in fields of prime characteristic, Manuscripta Math. 59 (1987), 457-469.
Garcia, A. and Voloch, J. F.: Duality for projective curves, Bol. Soc. Brazil. Mat (N.S.) 21 (1991), 159-175.
van der Geer, G. and van der Vlugt, M.: Generalized Reed-Müller codes and curves with many points, preprint.
Hefez, A.: Non-reflexive curves, Compositio Math. 69 (1989), 3-35.
Hefez, A. and Kakuta, N.: Tangent envelopes of higher order duals of projective curves, Rend. Circ. Mat. Palermo, Suppl. 51 (1998), 47-56.
Hefez, A. and Voloch, J. F.: Frobenius non classical curves, Arch. Math. 54 (1990), 263-273.
Hirschfeld, J. W. P. and Thas, J. A.: General Galois Geometries, Oxford University Press, Oxford, 1991.
Homma, M.: Space curves with degenerate strict duals, Comm. Algebra 20 (1992), 867-874.
Kaji, H.: Strangeness of higher order space curves, Comm. Algebra 20 (1992), 1535-1548.
Lang, S.: Abelian Varieties, Interscience, New York, 1959.
Rathmann, J.: The uniform position principle for curves in characteristic p, Math. Ann. 276 (1987), 565-579.
Rück H. G. and Stichtenoth, H.: A characterization of Hermitian function fields over finite fields, J. Reine Angew. Math. 457 (1994), 185-188.
Segre, B.: Forme e geometrie Hermitiane, con particolare riguardo al caso finite, Ann. Mat. Pura Appl. 70 (1965), 1-201.
Stöhr, K. O. and Voloch, J. F.: Weierstrass points and curves over finite fields, Proc. London Math. Soc. 52 (1986), 1-19.
Tate, J.: Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134-144.
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Korchmáros, G., Torres, F. Embedding of a Maximal Curve in a Hermitian Variety. Compositio Mathematica 128, 95–113 (2001). https://doi.org/10.1023/A:1017553432375
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DOI: https://doi.org/10.1023/A:1017553432375